Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
Series: Department of Mathematics, University of Maryland,
College Park
Adviser: L Greenberg
458
Peter Walters
Ergodic Theory -
Introductory Lectures
Springer-Verlag
Berlin • Heidelberg • New York 1975

Dr. Peter Walters
Mathematics Institute
University of Warwick
Coventry/England
Library of Congress Cataloging in Publication Data
Walters, Peter, 1943-
Ergodlc theory.
(Lecture notes in mathematics ; 458)
Bibliography: p.
Includes index,
1. Ergodic theory. I. Title. II. Series:
Lectures note's in mathematics (Berlin) ; 458.
QA3.L28 no. 458 ,QA313] 510'.8s c515'.42] 75-9853
ISBN O-387-O7163-6
AMS Subject Classifications (1970): 28A65
ISBN 3-540-07163-6 Springer-Verlag Berlin - Heidelberg • New York
ISBN 0-387-07163-6 Springer-Verlag New York ■ Heidelberg ■ Berlin
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Preface
These are notes of a one-semester introductory course on Ergodic
Theory that I gave at the University of Maryland in College Park
during the fall of 1970. I assumed the audience had no previous
knowledge of Ergodic Theory. My aim was to present some of the basic
facts in measure theoretic Ergodic Theory and Topological Dynamics and
show how they are related so that the audience would have the
foundations to read the research papers if they wished to pursue the subject
further.
At the beginning of Chapter 1 I give a list of examples of
measure-preserving transformations and at the end of each section of
Chapter 1 I investigate whether these examples have the properties
discussed in that section. These examples were chosen because of
their varied properties and importance in the subject. Similarly in
Chapter 5, on Topological Dynamics, a list of examples is given and
the properties discussed in that chapter are considered for these
examples.
I tried to deal with entropy as simply as possible. In the
discussion of entropy I have inserted without proof some of the more
difficult theorems when I thought they were relevant to the discussion.
In particular I have discussed the recent deep results of D. S.
Ornstein on Bernoulli automorphisms and Kolmogorov automorphisms.
In the final chapter I have presented the new treatment of
topological entropy due to R. E. Bowen. One of the beauties of this
treatment is that topological entropy can be defined for a uniformly
continuous map of any metric space and that its value remains
unchanged under certain types of covering maps. This enables one to
give an elegant calculation of the topological and (Haar) measure

IV
theoretic entropies of affine transformations of finite-dimensional
tori.
Since these notes have not been fully edited many references are
missing and it is likely that credit is often not given where it is
due. The theorems and definitions are numbered independently, but a
corollary is given the same number as the theorem to which it is a
corollary.
Thanks are due to Victor Charles Stasio and Suellen Eslinger who
took notes of the course and also to Allan Jaworski for editing and
compiling the bibliography. Special thanks are due to Betty Vander-
slice for her superb typing.
—Peter Walters

Contents
Chapter 0 : Preliminaries 1
§1. Introduction 1
§2. Measure Theory 3
§3. Hilbert Spaces 8
§4. 'Haar Measure 9
§5. Character Theory 10
§6. Endomorphisms of Tori 12
Chapter 1: Measure-Preserving Transformations 16
§1. Examples 16
§2. Problems in Measure Theoretic Ergodic Theory 19
§3. Recurrence 20
§4. Ergodicity 21
§5. The Ergodic Theorem 29
§6. Mixing 37
Chapter 2: Isomorphism and Spectral Invariants 51
§1. Isomorphism of Measure-Preserving Transformations 51
§2. Conjugacy of Measure-Preserving Transformations 53
§3. Spectral Isomorphism 54
§4. Spectral Invariants 57
§5. Examples 59
Chapter 3 : Measure-Preserving Tran s format ions 6 3
with Pure Point Spectrum
§1. Eigenfunctions 63
§2. Pure Point Spectrum 64
§3. Group Rotations 67
Chapter 4: Entropy 70
§1. Partitions and Subalgebras 70
§2. Entropy 72
§3. Conditional Entropy 76
§4. Properties of h(T,A) 80
§5. Properties of h(T) 83
§6. Examples 92
§7. How good an invariant is entropy? 9 6
§8. Bernoulli and Kolmogorov Automorphisms 9 8
§9. Pinsker Algebra 107
§10. Sequence Entropy 108
§11. Comments 109
§12. Non-invertible Transformations 110

V!
Chapter 5: Topological Dynamics 112
§0. Introduction 112
§1. Minimality 113
§2. Topological Transitivity 117
§3. Topological Conjugacy and Discrete Spectrum 122
§4. Invariant Measures for Homeomorphisms 128
Chapter 6: Topological Entropy 140
§1. Definition by Open Covers 140
§2. Bowen's Definition 146
§3. Connections with Measure Theoretic Entropy 155
§4. Topological Entropy of Linear Maps
and Toral Affines 160
§5. Expansive Homeomorphisms 168
§6. Examples 182
Bibliography 18 5
Index
197

Chapter 0 ; Preliminaries
§1. Introduction
Generally speaking, ergodic theory is the study of
transformations and flows from the point of view of recurrence properties,
mixing properties, and other global, dynamical properties connected
with asymptotic behavior. Abstractly,'one has a space X and a
transformation T of X (or a family {T : t € R} of
transformations of X) with some structure on X which is preserved by T
(or by {T }). The nature of most of the work so far can be
categorized into one of the four following types:
(1) measure theoretic :
Here one deals with a measure space X and a measure
preserving transformation T: X •* X.
(2) topological:
Here X is a topological space and T: X -* X is a continuous
map.
(3) mixture of (1) and (2):
In this situation X is a topological space equipped with a
measure m on its Borel sets while T: X -* X is continuous and
preserves m.
(4) smooth:
One considers a smooth manifold X and a smooth map
T: X -> X.
We shall deal with some topics from (1), (2), and (3).
To see how this study arose consider, for example, a system of
k particles in 3-space moving under known forces. Suppose that the
phase of the system at a given time is completely determined by the

2
positions and the momenta of each of the k particles. Thus, at a
given time the system is determined by a point in 6k-dimensional
space. As time continues the phase of the system alters according to
the differential equations governing the motion, e.g., Hamilton's
equations
dqi 9H dpi 9H
dt 3p. dt 3q.
If we are given an initial condition and such equations can be
uniquely solved then the corresponding solution gives us the entire history
of the system, which is determined by a curve in phase space.
If x is a point in phase space representing the system at a
time tQ, let T (x) denote the point of phase space representing
the system at time "t+tn- From this we see that T. is a
transformation of phase space and, moreover, Tn = id., T., = T.°T . Thus
r 0 t+s t s
{T.: t € R} is a one-parameter group of transformations of phase
space. In dynamics one is interested in the asymptotic properties of
the family {T }. It seems reasonable to study the system at discrete
times tn, 2tQ, 3tQ,..., i.e., study the family {T^ } , , since we
expect the properties of {T } to be reflected in those of {T^ }.
r r0
For this reason, as well as the fact that it is simpler, one
studies individual transformations and their iterates. One is
particularly interested in the flow on an energy surface, which is
sometimes smooth (hence considerations of type (4) arise), and sometimes
is not (one then investigates along the lines of (2) ). Measure
theory enters the picture via a theorem of Liouville which tells us
that if the forces are of a certain type one can choose coordinates
in phase space so that the usual 6k-dimensional measure in these
coordinates is preserved by each transformation T..
Around 1900 Gibbs suggested using the measure-theoretic approach

3
in mechanics because of the difficulty of solving the equations of
motion and also because this deterministic approach does not answer
several important questions in mechanics. In discovering statistical
mechanics Gibbs suggested looking at what happens to subsets of phase
space. For example, if A and B are subsets of phase space what
is the probability that the system is in B at the time t given
that the system is in A at the time t-.? Given that the system
begins in A at time tQ what is the average time the system spends
in B? Such questions motivate the type of study undertaken in
ergodic theory.
We now list some general references for the material we shall
discuss :
For topics of types (1) and (3) mentioned above see Halmos [2],
Billingsley [1], Hopf [1], Jacobs [1] [2], Parry [3], Rohlin [3][4][5],
Friedman [1], Shields [2]. In addition to the Shields notes, further
details on the results of Ornstein described in Chapter 4 may be found
in a forthcoming book by Friedman and Ornstein [2].
For material of type (2) see Gottschalk and Hedlund [1],
Nemytskii and Stepanov [1], and Ellis [1].
And material of type (i+) may be found in Avez and Arnold [1],
Smale [1], Abraham [1], Abraham and Robbin [1], Nitecki [1].
Khinchin [1] provides a good sketch of the foundations of ergodic
theory. For extensive bibliographies see Jacobs [1][2], Gottschalk
[1], and Smale [1]. A recent survey is Mackey [1].
§2. Measure Theory
General reference - Halmos [1].
We recall some fundamental notions from measure theory.
Let X be a set. A a-algebra of subsets of X is a collection
B of subsets of X satisfying:

4
(1) X € B, (2) B € B = X\B € B,
(3) B € B, n > 1 = (J B € B.
n , n
n=l
We then call (X,B) a measurable space. A measure space is a
triple (X,B,m) where X is a set, B is a a-algebra of subsets
of X, and m is a function m: B •*■ R satisfying
m( U B ) = E m(B )
n=l n n=l n
if {B } is a pairwise disjoint sequence of elements of B. We say
that (X,B,m) is a probability space, or a normalized measure space,
if m(X) = 1. We shall usually deal with such spaces.
A collection A of subsets of a set X is an algebra if:
(1) X € A, (2) A € A = X\A € A,
n
(3) A ,...,An € A = U A. € A.
x n i=l x
When one is trying to equip a measurable space (X,B) with a
measure one usually knows what the measure should be on an algebra
A £. 8, and so, one would like to know when this function defined
on A can be extended to a measure on B. The following result
deals with this situation.
Hahn-Kolmogorov Extension Theorem:
Given a set X, an algebra A of subsets of X, let m: A ■* R
be a function satisfying
m(X) = 1, m((jAn) = Lm(An)
n n
whenever A € A V n, I J A € A, and the (A } are disjoint. Then
n
there is a unique probability measure m defined on the a-algebra
generated by A such that m(A) = m(A) whenever A € A.

5
A monotone class of subsets of X is a collection C of subsets
of X such that if E, c E„ c ... belong to C then U E € C and
n
if F. D F, D ... belong to C then (~) F € C.
n
Theorem:
If A is an algebra of subsets of X then the a-algebra
generated by A equals the monotone class generated by A.
If (X,B,m) is a finite measure space, one can easily deduce
from this theorem that if A is an algebra generating the a-algebra B
then for any B € B and e > 0 there exists A € A with m(AAB) < e.
(In fact,
C = {B€B|Ve>0 ] B € A with m(BAB ) < e}
is a monotone class and contains A.)
Notation: If A is an algebra we shall write a(A) for the
a-algebra generated by A.
Direct Products :
Let (X.,8.,m.), i € Z be probability spaces. Their direct
product
(X,B,m) = TT (Xi,Bi,mi)
i=-~
is defined as follows:
(a) X = U X±
(b) Let n, < n„ < ... < n be integers, and A € 8 i = 1,...,r.
i i
We define a measurable rectangle to be a set of the form
{ (x. ) € X: x € A V i: 1 S i S r} .
1 n. n.
J il
Let A be the collection of all finite unions of measurable
rectangles. A is an algebra: (1) and (3) are obvious; to show (2)

6
observe that
r
X\{(x.) |x €A , lïiîr} = U {(x.) | xn € Xn \ A } € À
a ni ni i=i -i ni ni ni
and that A is closed under finite intersections. Let B be the
a-algebra generated by A.
(c) Each element of A can be written as a disjoint finite union of
measurable rectangles so that we define
r
m({(x.) | xn_ €A , lsisr}) = J[ mn (A )
J i i i=l i i
and then extend m to A in the obvious manner. The conditions of
the Hahn-Kolmogorov Theorem can be shown to be satisfied, and thus we
can extend m to B. Hence, we obtain a probability space.
Measurable Functions :
A Borel subset of R is a member of the a-algebra generated by
the open sets.
Let (X,B,|j.) be a measure space; f: X -* R is measurable if for
all c € R, f~ (c,-) € B, or equivalently f~ (D) € B V Borel sets
D c R.
A function f: X -* C, f = f. + if„ is measurable if f,: X -* R
and f„ : X -* R are measurable.
If X is a topological space and B is the a-algebra of Borel
subsets of X (the a-algebra generated by the open subsets of X)
then each continuous function f: X -* C is measurable.
Integration :
n
A simple function is a function of the form ]T a.X. , where
i=l 1 i
a. € R, the {A.} are disjoint members of B, and X denote
s
l
the characteristic function of A.. Simple functions are measurable.
We define the integral for simple functions by:
(• n n
( E a.X. )dm = E a.mCAO.
J i=l 1 Ai i=l 1 1

7
Suppose f: X -* R is measurable and f > 0; then there exists
an increasing sequence of simple functions f j> f. For example we
could take
fn(x)
i-1 ■ .c i-1 «- et \ i • t ^ri
if S f(x) < — i=l,.. . ,n2
2n 2n
In if f(x) > n.
We define / fdm = lim / f dm and note that this definition is înde-
* n
pendent of the chosen sequence {f }.
Suppose f: X •*■ R is measurable; then f = f - f where
f+(x) = max {f(x),0} > 0
f"(x) = max {-f(x),0} 5r 0.
We say that f is integrable if / f dm, / f~dm < », and we define
[ fdm = I f+dm - [ f"dm.
Now if f: X -* C is measurable, f = f, + if„, f is integrable if
f, and f„ are integrable and we define
I fdm = I fxdm + i I
dm + i | f„dm.
Observe that f is integrable if and only if |f| is integrable.
Let L (m.) denote the class of all integrable functions (X,B,|j.) -* C.
Lebesgue Dominated Convergence Theorem :
Suppose {f } is a sequence of measurable functions, f -*f a.e.
and there exists an integrable function g such that |f (x)| s g(x)
a.e. for all n; then f and each f are integrable and
f fn^ - } fd,

§3. Hilbert Spaces
General reference - Halmos [3].
H is a Hilbert space if it is a Banach inner product space,
i.e., (1) H is a vector space over the complex numbers C
(2) There is an inner product on H, i.e., a map
(•,•): H x H -* C such that :
(a) (•,■) is bilinear
(b) (f,f) ï 0 for all f in H
(c) (f,f) = 0 if and only if f = 0
(d) (f,g) = (g,f).
(3) ||f|| = (f,f) is a norm on H inducing a complete metric
topology on H.
Let (X,B,m) be a measure space. Consider all measurable func-
tions f: X •*■ C such that / |f| dm < »; we define an equivalence
relation on this set by saying that f ~ g if and only if f = g a.e.
The set of equivalence classes relative to this relation forms a Hil-
2 2
bert space which we denote by L (X,B,m) = L (m), where the inner
product of two functions is given by
(f,g) = j fgdm.
Recall: The Schwarz Inequality
In any Hilbert space H
|(f,g)| S ||f||-|lgll for all f,g in H.
A unitary operator U on a Hilbert space H is an isomorphism
of H, i.e., U is a linear bijective map preserving the inner
product ((Ux,Uy) = (x,y), x,y in H). It follows that U is
continuous .

9
§4. Haar Measure
General reference - Pontrjagin [1].
Theorem:
Let G be a compact topological group; then there exists a
finite measure m defined on the Borel subsets of G such that
m(xE) = m(E) for all x in G, and for all Borel sets E. We call
such a measure a Haar measure.
E.g., let K = {z € C: |z| = 1} and m denote normalized
circular Lebesgue measure. Then m(aU) = m(U) for all sets U measurable
on K.
Theorem;
If m and p. are both finite Haar measures on the compact
topological group G then m = cm. where c > 0. Thus there exists
only one normalized Haar measure on G.
Remarks :
(1) If U c G is a non-empty open set then it has positive Haar
measure. This is because
G = U gD=g,U0g,U0.,.0gU
g€G X n
by compactness.
2
(2) In the Hilbert space L (K,B,m) where m is the Haar measure on
the unit circle K, the functions
f(z) = z11, n € Z
form an orthonormal basis.

10
§5. Character Theory
General reference - Pontrjagin [1].
Many of our examples will be rotations, endomorphisms or affine
transformations of compact groups. (We mean endomorphism in the sense
of topological groups, i.e., an abstract group endomorphism v;hich is
continuous.) In some proofs we will use the character theory of
compact abelian groups, which we summarize in this section. For those
not familiar with character theory, proofs in the later sections
involving characters will usually be preceded by the proof in a special
case where the group used is the unit circle and then classical
Fourier analysis will be used.
Let G be a locally compact abelian group with a countable
topological base. Let G denote the collection of all continuous homo-
morphisms of G into the unit circle K. The members of G are the
characters of G, G is an abelian group under the operation of
pointwise multiplication of functions. With the compact open topology
G becomes a locally compact abelian group with a countable
topological base. We have the following results:
(1) G is compact «• G is discrete.
(2) (G) is naturally isomorphic (as a topological group) to G, the
isomorphism being given by the map
a ■* a where a(y) = y(a) for all y € G.
(3) If G is compact then G is connected «» G is torsion free.
(4) G, xG„ = G, xG„ where "x" denotes direct product.
So in some sense we can study compact abelian groups by studying
discrete countable groups.
Examples: (See §6 for proofs.)
(i) Let G = K = {z € C: |z| =1}. Each homomorphism of K to
itself is of the form z -* z11 (n € Z) so that G ^ Z.

11
(ii) Let G = Kn the n-torus. By (1+) Kn ^ Zn and in fact each
member of Kn has the form
f\ z-]_ ) z2 ' • * • ' zn' ~ z]_ z2 *"* n
where (p ,...,p ) € Zn.
(5) If H is a closed subgroup of G and H i G there exists a
Y i G, y * 1 such that y(h) =1 V h € H. (We shall write this
Y(H) = 1.)
(6) Let G be compact. The members of G are mutually orthogonal
2
members of L (m), where m is Haar measure.
Proof. It suffices to show
Y(x)dm(x) = 0 if Y ¥ !•
If a € G then since m is Haar measure
Y(x)dm(x)
Y ( a.x ) dm ( x ) = Y(a)
Y(x)dm(x)
Choosing a so that y(,&) i 1 we have Y(x)dm(>0 = 0. //
(7) If G is compact, the members of G form an orthonormal basis
o
for L (m) where m is normalized Haar measure.
This is part of the Peter-Weyl theorem and can be easily deduced
from the Stone-Weierstrass theorem, which implies that finite linear
combinations of characters are dense in C(G) = space of complex-
valued continuous functions of G.
(8) If A: G -* G is an endomorphism we can define an endomorphism
A: G -* G by Ay = Y°A, y i G. It is easy to see that A is one-
to-one if and only if A is onto and A is onto if and only if A
is one-to-one. Therefore A is an automorphism if and only if A
is an automorphism.
Recall that for compact groups G, G is metric iff G has a

12
countable topological base.
§6. Endomorphisms of Tori
We shall view the n-torus in two ways:- multiplicatively as
Kn = KxKx...xK , and additively as Rn/Zn where Rn is n-space and
n times
Z is the subgroup of R consisting of points with integer
coordinates. A topological group isomorphism is given by K -* R /Z ,
(e
2rrix, 2Trix_ , „ n
1,...,e n) I* (Xl,...,xn) + Zn.
Theorem:
(1) Every closed subgroup of K is either K or is a finite cyclic
group consisting of all p-th roots of unity for some integer p > 0.
(2) The only automorphisms of K are the identity and the map
z K- z
(3) The only homomorphisms of K are the maps
4> : z h- z , n € Z.
(4) The only homomorphisms of K to K are maps of the form
(z, ,...,z ) >->■ z, •...•z where m, , . . . ,m_ € Z.
In 1 n in
Proof: (1) Let H be a closed subgroup of K; if H is
infinite it has a limit point so, V e > 0 3 a,b € H 3 d(a,b) < e,
a t b. Then d(b~ a,l) < e and therefore the elements of H are
e-dense in K V e. Thus, H = K.
If H is finite and has p elements then a^ = 1 V a € H. So
each element of H is a p-th root of unity, and since there are p
elements in H, H must consist of all the p-th roots of unity.
(2) Let 9: K -* K be an automorphism. 9(1) = 1. Since -1 is the
only element of K of order 2 we have 9(-l) = -1. Since i,-i

13
are the only elements of order 4 either 9(i) = i and 9(-i) = -i
or 9(i) = -i and 9(-i) = i. Consider the first case. Since 9
maps intervals to intervals, the interval Cl,i] from 1 to i is
either mapped to itself or to [i,l] (all intervals go anticlockwise).
But since [l,i] does not contain -1 it cannot be mapped to Ci,l]
so 9[l,i] = [l,i]. The only element of order 8 in Cl,i] is
s and so this must be fixed by 9. Therefore gCl^"1 ]
:—>■ . k . k
Cl,eTrl/1+]. By induction one shows that 9(e2Trl/2 ) = e2Tri/2 for
v
each k > 0. It follows that 9 fixes all the 2 -th roots of unity
V k > 0 and hence is the identity. In the second case one shows
k . k
that 9(e2Trl/2 ) = e"2Tri/2 V k > 0 and hence 9(z) = z-1, z € K.
(3) Let 9: K -* K be an endomorphism. If 9 is non-trivial, its
image, 9(K), is a closed connected subgroup of K and so 9(K) = K
by (1). The kernel Ker 9 is a closed subgroup of K so either
Ker 9 = K or Ker 9 = H , the group of all p-th roots of unity, for
some p. The first case corresponds to trivial 9. If Ker 9 = H
let a : K/H •* K be the isomorphism given by a (zH ) = zP, and let
9,: K/H -* K be the isomorphism induced by 9 (9,(zH ) = 9(z) ).
1 p r J 1 p
Then 9]_aZ is an automorphism of K and by (2) either ®-\aZ (z) = z
V z € K or 9,a-1(z) = z-1 V z € K. Hence either 9(z) = 9,(zH ) =
1 p 1 p
8,a"1(zP) = zP V z € K or e(z) = z~P V z € K.
1 P
(4) Let Y..-: K -► Kn be defined by y-Cz) = (1,1, . . . ,l,z ,1, . . . ,1).
i-th place
If 6: K -* K is a homomorphism then 9oy-: K -* K is an endomorphism
m.
and so eoY^(z) = z 1 for some m. € Z by (3). Hence
e(z1,...,zn) = e(Y1(z1)-Y2(z2)-...-Yn(zn))
= 9y1(z1)-9y2(z2)'-• •'9Yn(zn)
m m, m_
= z/-z,2'....znn . //
1 l n

14
Theorem:
(1) Every endomorphism A: K -* K is of the form:
a a a a
ac > ,11 in m nn,
A(z1,...,zn) = (z1 -...-zn ,...,z1 -...-zn )
where a.- € Z. In additive notation,
*.»)-^(T
n
(2) A maps Kn onto Kn iff detCa..] i 0.
(3) A is an automorphism of Kn iff det[a..] = ±1.
Proof: (1) Let tt . : K -* K be the projection to the i-th
coordinate. Then tt.oA: Kn -* K is a homomorphism, so by (i+) of the
previous theorem
rr.oACz,,...^) - ^■^■...■^
where a.. € Z.
(2) Assume detCa..] =0. 3 nu ,. . . ,m integers not all zero 3
m,A, + ... +m A =0 where A. is the i-th row of A. Then each
11 n n l
n mi mn
point (<u,,...,<u ) of K in the image of A satisfies co, ...m =1.
Thus A(Kn) t Kn. Conversely if A(Kn) t Kn then the points of
A(K ) are annihilated by a nontrivial character of K , say
m m
(z1,...,zn) -> zx -...-znn (this is by (5) of §5). Then
mnA, + . . . +m A =0 and so det[a. - ] = 0 .
il n n i]
(3) If A is an automorphism represented by a matrix [A] then A
is also an automorphism represented by a matrix [B], and since AA =
I = A A we have that [B] = [A]~ . Since [B] is an integer matrix,
det[A] = ±1. Conversely, if det[A] = ±1, [A]~ has integer entries
and if B is the endomorphism of Kn it defines we have AB = BA = I. //

15
Notation :
If A is an endomorphism of the n-torus, [A] will always
denote the associated matrix and A will denote the linear
transformation of Rn determined by [A]. So if tt : Rn -* Rn/Zn is the natural
projection (tt(x) = x + Z ) we have ttA = Att .
Let A: K -* K be an endomorphism. We now consider how the map
A: K -* Kn (introduced in §5) acts as a map of Zn when Kn is
identified with Z by the isomorphism:
2. .- n
when y(z1)z2'-••>zn) = z±1'z2~'' " 'zn
One readily checks that the endomorphism À: Z -* Z is given by
CA]t
,mls
n
where [A] denotes the transpose of the matrix [A].

Chapter 1 : Measure-Preserving Tran s fo rmat ions
§1. Examples
Suppose (X,,8.,m,), (X„,B„,m„) are probability spaces.
Definition 1.1:
a) T: X1 ■* X„ is measurable if T-1(B2) c %1 (i.e.,
B2 € B2 = T-1B2 € B1).
b) T: X, ->■ X2 is measure-preserving if T is measurable and
m1(T"1(B2)) = m2(B2) V B2 € Bj.
c) We say that T: X, -* X„ is an invertible measure-pre serving
transformation if T is measure-preserving, bijective, and T~ is
also measure-preserving.
Remarks :
(1) We should write T: (X. ,B, ,m., ) -* (X2,B2,m„) since the measure-
preserving property depends on the B's and m's.
(2) If T: X, -* X2 and S: X, -» X, are measure-preserving so is
SoT: Xx - X3.
(3) Measure-preserving transformations are the structure preserving
maps (morphisms) between measure spaces.
(4) We shall be mainly interested in the case (X.,B,,nu) = (X2,B2,m2)
since we wish to study T (see §1, Ch. 0).
In practice it would be difficult to check, using Defs. 1.1,
whether a given transformation is measure-preserving or not since one
usually does not have explicit knowledge of all the members of B.
However we often do have explicit knowledge of an algebra A
generating B (for example, when X is the unit interval A may be all
finite unions of intervals, and when X is a direct product space

17
A may be the collection of all finite unions of measurable
rectangles). The following result is therefore desirable in checking
whether transformations are measure-preserving or not.
Theorem 1.1:
Suppose (X-, ,8..,m.. ), (X2,B2,m„) are probability spaces and
T: X, ■* X„ is a map. Let A„ be an algebra which generates B„.
If A2 € A2 » T_1(A2) € B1 and m1(T-1(A2)) = m2(A2) then T is
measure-preserving.
Proof: Let C2 = {B€B2: T_1(B) €B , m1(T-1(B)) =m2(B)}; we
want to show that C„ = B„. However A„ £. C„ and C„ is easily
seen to be a monotone class, so the result follows since the a-algebra
generated by C„ is the monotone class generated by A2. //
Examples of Measure-Preserving Transformations :
(1) I = identity on (X,B,m) is obviously measure-preserving.
(2) Let K = {z€C: |z| =1}, 8= Borel sets, and m = Haar measure.
Define T:K-*K by T(z)=az where a is a fixed element of K.
T is measure-preserving since m is Haar measure.
(3) The transformation T(x) = ax defined on any compact group G
(where a is a fixed element of G) preserves Haar measure.
(4) Any continuous endomorphism of a compact group onto itself
preserves Haar measure.
Proof: Let A: G -* G be a continuous endomorphism, and
m = Haar measure on G. Let |x(E) = m(A~ (E)). p. is a Borel
probability measure and
M-(Ax-E) = m(A_1(Ax-E)) = m(x-A_1E) = m.(E).
Since A maps G onto G, \j. - m by the uniqueness property of Haar
measure. //
For example, T(z) =z preserves Haar measure on the unit circle.

18
(5) Any affine transformation of a compact group G preserves Haar
measure. An affine transformation is a map of the form T(x) = a-A(x)
where a € G is fixed and A: G -* G is a surjective endomorphism.
T is measure-preserving because it is the composition of measure-
preserving transformations. When A = I we have example (3) and
when a is the identity element of G we have example (4).
(6) Let Y = {0,...,k-l}, and give measure p. to i such that
k-1
Y. P- - I- We let X = Tf Y together with the direct product
i=0 1
measure. Define T: X -* X by:
T({xi>) = {yi>, where yi = xi+1-
T preserves the measure of each measurable rectangle and thus it
preserves the measure of sets which are finite disjoint unions of
measurable rectangles. By Theorem 1.1 T is measure-preserving. We call
T the two-sided (p..,...,p. ..)-shift.
(7) Let Y be as above, X = IT Y with the direct product measure.
0
Let T: X -> X be defined by
(xQ,x1, . . .) h- (x1,x2, . . . ) .
By an analogous argument to the one in example (6) we see that T is
measure-preserving. We call T the one-sided (pQ,...,p, .)-shift■
2
(8) Let I be the unit square equipped with Lebesgue measure and I
2
the unit interval with Lebesgue measure. Then p: I -* I defined by
p(x,y) = x is measure-preserving.
Given any set X,, any probability space (X„,B„,m2) and any
onto
map T: X. -* X„ we can choose a a-algebra 8. and a measure m1
on X-, to make T measure-preserving. In fact let 8, = T 8„ and
define m, by nuCT B„) = nuCB,).
Conversely, if (X, ,B., ,m., ) is any probability space, X2 any set

19
onto
and T: X, -* X„ any map, then we can choose a a-algebra B„ and a
sure m„ on X„ so that T is measure-preserving.
Put
B2 = {B: B c x2 and T-1B € B1>
and m2(B) = m1(T~1B) for B € B2-
§2. Problems in Measure Theoretic Ergodic Theory
(a) External Problems :
How do we apply measure theoretic ergodic theory to other
branches of mathematics and physics? In these applications one has
a space X with some structure on it, and a map T of X which
preserves this structure. To apply the theory of measure preserving
transformations one needs an invariant measure for T which acts
"nicely" with respect to the structure on X. For example, if X is
a topological space we would like the measure to be a Borel measure
which is positive on non-empty open sets.
Examples :
(1) Hamiltonian Mechanics : Here one has a one-parameter group of
diffeomorphisms of a manifold and there is a smooth measure on the
manifold preserved by each diffeomorphism.
(2) Number Theory: To study continued fractions one studies
T: [0,1) -> [0,1) given by:
f 0 if x = 0
T(x) = <
[{1/x} if x ?! 0}
where the {y} denotes the fractional part of y. T preserves the
Gauss measure on [0,1) which is given by:
m(A) =
lo
-— [ — dx, A = [0,1).
g 2 JA 1+x

20
(b) Internal Problems :
The main internal problem in measure theoretic ergodic theory is:
Given two measure preserving transformations when are they isomorphic?
(i.e., when can we consider them to be the same?) We look for
invariants. A property P is an invariant if when T, has the property P
and T„ is isomorphic to T, then T„ has the property P.
Invariants give good negative answers, i.e., if T, has property P and
T„ does not, then T, and T„ are not isomorphic. The invariants
we shall study are of two types:—spectral invariants and entropy.
Before discussing the notion of isomorphism we shall introduce
some general concepts such as recurrence and mixing.
§3. Recurrence
Theorem 1.2: (Poincaré Recurrence Theorem)
Let T be a measure-preserving transformation of a probability
space (X,B,m). Let E € B, m(E) > 0. Then almost all points of E
return infinitely often to E under positive iteration by T, (in
fact, we have the stronger result that: 3 F c E, m(F) = m(E) 3 if
x € F 3 integers 0 < n1 < n2 ... 3 T -"-(x) € F V i).
oo
Proof: For N =: 0 let EN = U T~n(E). We have T-1(EN) = EN+1,
n=N
EN C EN-1 C EN-2 C ■•• C V E C V and m(EN+l) ^(T_1(EN)) = m(EN)
since T is measure-preserving. Therefore, for each N, m(EN) =
m(EQ) and,
m( D EM) = m(En).
N=0 N u
O EM = 0 U T~nE, which is the set of all points entering E
N=0 N N=0 n=N
infinitely often under positive iteration by T. Moreover
F = E n ( O EM) consists of all points of E which enter E
N=0 N

21
infinitely often under positive iterates of T. Since O EM c En
N=0 N U
and both sets have the same measure,
m(F) = m(E fl 0 E ) = m(E n E ) = m(E).
N=0 N u
It remains to show a point of F returns to F infinitely often.
n ■
Let x € F, then 3 0 < n, < n ... 3 T -"-(x) € E V l. Consider
T '(x). T '(x) € E and enters E infinitely often under positive
iterates, namely, n„-n,, n.-n,, ... since T -1 l (T x(x))=T 1(x) €E;
n, n-;
thus T x(x) € F. Similarly one shows T (x) € F V l. //
Remark:
We do not need that T be measure-preserving in the hypothesis;
we need only assume that T is incompressible, i.e., if B € B,
T-1B c B then m(B) = m(T-1B).
§4. Ergodicity
Let (X,B,m) be a probability space and T: X -* X be a measure-
preserving transformation. If T~ B = B for B € B, then
T~ (X\B) = X\B and we could study T in two separate parts,
namely T|B and tIv\r- If ° < m(B) < 1 this has simplified the
study of T. We need a concept of irreducibility for measure-
preserving transformations, such that if T has this irreducibility
property then the study of T cannot be split into two parts as
above. Ergodicity is such a concept. Also, we would like some way
of splitting a measure-preserving transformation into ergodic parts
in a canonical way. This can be done in reasonably well behaved
measure spaces. (See Rohlin [3].)
Definition 1.2:
T: (X,B,m) -* (X,B,m) is ergodic if for B € B, T-1B = B =»m(B) = 0
or m(B) = 1.

22
Theorem 1.3:
The following are equivalent for measure-preserving T: X -* X:
(1) T is ergodic.
(2) m(T-1BAB) = 0, B € B => m(B) = 0 or 1.
(3) V A,B € B, m(A),m(B) > 0 3 n > 0 3 m(T~nA fl B) > 0.
Proof: (1) => (2). Suppose m(T-1BAB) = 0. Let
oo oo .
B_ = O U t~1b « B- Then
n=0 i=n
B_ = (BO T-1B U T~2B U . . . ) fl (T-1B U T~2B U . . . ) fl (T~2B U T~3B U . . . ) fl . . . ;
and therefore T~ B_ = B_ and m(B_) = m(B). Hence, by (1),
m(B_) =0 or 1, and therefore m(B) = 0 or 1.
(2) =• (3). Let m(A) > 0, m(B) > 0 and suppose (3) is false,
i.e., V n > 0 m(T-nAriB) = 0. Then
m(( U T"nA) fl B) =0.
n = l
oo
Let A' = U T~nA. Then T-1A' c a' and m(A') = m(T A'); so that
n=l
m(T-1A'AA') = 0, and by (2) we have m(A') = 0 or 1. But T-1AcA'
and T is measure-preserving, so that m(A ) = 1. But this
contradicts the above fact that m(A' (IB) = 0.
(3) => (1). Suppose (1) is false, i.e., 3 B € B, T-1B = B and
0 < m(B) < 1. Then m(T~nB fl (X\B)) =0 V n > 0 which contradicts
(3). //
A characterization of ergodicity in terms of functions is given
by the following results.

23
Theorem 1.4:
Let T: (X,8,m) -* (X,B,m) be measure-preserving; then the
following are equivalent:
(1) T is ergodic.
(2) Whenever f is measurable and (foT)(x) = f(x) V x € X then
f is constant a.e.
(3) Whenever f is measurable and (foT)(x) = f(x) a.e. then f is
constant a.e.
(4) Whenever f € L2(m) and (f°T)(x) = f(x) V x € X then f is
constant a.e.
(5) Whenever f € L2(m) and (f°T)(x) = f(x) a.e. then f is
constant a.e.
Proof: (1) => (3). Suppose f is measurable and foT = f a.e.;
while T is ergodic. We can assume that f is real-valued for if
f is complex-valued we can consider the real and imaginary parts
separately. Define
X(k,n) = {x: k/2n £ f(x) < (k+l)/2n} k € Z, n > 0.
We have
T-1X(k,n)AX(k,n) c {x: (foT)(x) i f(x)}
and hence m(T~ X(k,n)AX(k,n)) = 0 so that by (2) of Theorem 1.3
m(X(k,n)) =0 or 1.
Fix n, then \J X(k,n) = X which is a disjoint union; so for
k€Z
each n 3 unique k 3 m(X(k ,n)) = 1. Let Y = (~) X(k ,n).
n n ' , n
n=l
Then m(Y) = 1 and f is constant on Y so that f is constant
a.e.
Trivially we have (3) => (2) => (4), (5) =» (4), and (3) =» (5).
So it remains to show:
(4) => (1). Suppose T-1E = E, E € B. Then X-, € L (m) and

24
(XE°T)(x) = X£(x) V x € X so, by (4) X£ is constant a.e. Hence
■f
m (E) = X^dm =0 or 1. //
Note:
A similar characterization in terms of L (m) functions is true,
since in the last part of the proof X„ is in L (m) as well as
2 12
L (m). Also we could use real L (m) or L (m) spaces.
Remark:
The following remark comes later (Theorem 5.5) but we preview it
now in order to analyze our examples.
Let X be a compact metric space and m a Borel probability
measure on X which gives positive measure to every non-empty open
set. If T: X -* X is continuous and ergodic with respect to m then
m({x | {Tnx | n 2: 0 is dense}}) = 1.
Proof: Let {U } -, be a base for the topology of X .
n'n=l
eo ea
ince
{Tnx |nî0} is dense in X ~ x € (~) U T~kU . S
n=l k=0 n
ea ea
T~ ( U T~ U ) c (_) T~ U and T is measure-preserving and ergodic
k=0 n k=0
ea ea
we have m( U T~ U ) = 0 or 1. Since U TU is a non-empty
k=0 n k=0 n
ea
open set we have m( \J T~ U ) = 1. The result follows. //
k=0 n
Note that this result is applicable when m is Haar measure on a
compact metric group and T is an affine transformation.
Examples :
We shall now see when the examples of § 1 are ergodic.
(1) I on (X,B,m) is ergodic iff all members of B have measure
0 or 1.
(2) Consider T: K -* K, T(z) = az. T is ergodic iff a is not a
root of unity.
Proof: Suppose a is a root of unity, then ap = 1 for some

25
p i 0. Let f(z) = zp; then clearly f i constant a.e. and f°T = f.
2
Conversely, suppose a is not a root of unity and foT =f, f €L (m).
ee
Let f(z)= ]T bz be its Fourier series. Then by above,
n=-~
E b anzn = E b z11 and therefore b (an-l) = 0. If n t 0 then
n n n
b =0, and so f is constant a.e. //
(3) Let T(x) = ax on a compact metric group G, then T is er-
godic iff {a } _ is dense in G. In particular, T ergodic =>
n cZi
G is abelian.
Proof: Suppose firstly that Tx = ax is ergodic. By the above
remark it follows that {a 'X-C]}7, is dense for some xn and so taI1}n
n-r n-r
is dense since if y € G 3 {n.} with a -"-x. -* yxn i.e. , a ± -* y.
Conversely, suppose {a } .„ is dense in G. This implies G is
abelian. Let f (. L2(m) and foT = f. By (7) of §5 of Chapter 0
f can be represented as 2_ b.y^) where y- i G. Then
T. b.Y-(a)r-Cx) = ^- b.y(x) so that if b. i 0 then y(a) = 1
i i
and so y. = 1. Therefore only the constant term of the Fourier
series of f can be non-zero, i.e., f is constant a.e. //
(4) For an endomorphism A of a compact metric group G necessary
and sufficient conditions for ergodicity are known. When G is
abelian, Halmos [4] and Rohlin independently proved that A is
ergodic «» whenever y°A = y> n > 0, then y = 1• Before proving
the general result we will illustrate the proof by showing that
2
A(z) = z on K is ergodic.
Suppose foA = f, f € L (m). We have that if f(z) = Y. a z"
n=-»
then 1- a z = *— a z11 and therefore a = a„ = a,, = . . . . So
n n n ^n 4n
ea
if n i 0 we must have a = 0 because 2_ la I < ». Therefore
n ' n '
— ea
f is constant a.e.

26
Proof of the general result : Suppose that whenever yAn = y we
have y = 1> and let f°A = f with f € L (m). Let f(x) have the
Fourier series *- a y where y € G and 2_ I a I < ». Then
n n n ' n '
£- a y where y € G and 2_ I a I
n n n ' n '
anYn(Ax) = 2_anYn(x)> so that if Yn > Yn°-A, Yn°A , ... are all
distinct their corresponding coefficients are equal and therefore
zero. So if a i 0, y_(A^) = y for some p > 0. Then y = 1 by
assumption and so f is constant a.e.
Conversely let A be ergodic and yA = y> n > 0. If n is
the least such integer, f = y +yA + .•. +yA is invariant under A
and not a.e. constant (being the sum of orthogonal functions),
contradicting the ergodicity. //
Consider now the case when G is the n-torus K . The ergodicity
condition then becomes: A: K -* K is ergodic «• the matrix [A] has
no roots of unity as eigenvalues.
Proof: Recall that, under the identification Kn ^ Zn, if
*!\
m /
n
€ Z then [A] m = Â(m). If A is not ergodic then
k k
[A] m = m for some m ?! 0 and k > 0. Then [A] has 1 as an
t - - - - t
eigenvalue so [A] has a k-th root of unity as an eigenvalue.
Conversely if [A] has a k-th root of unity as an eigenvalue
then [A] has 1 as an eigenvalue so that [A] - I induces a
singular linear transformation R . Hence [A] - I induces a many-
to-one map of Z into Z and so there exists 0_ i m € Z with
k
[A] m = m. //
So, for example, all the endomorphisms z-*z, |k|>l, of K
are ergodic.
Chu [1] has considered the case when G is nonabelian. He has
shown that a continuous endomorphism of a compact group G onto
itself is ergodic if and only if the induced map on the representation

27
ring R(G) has no finite orbit except the constant functions. The
representation ring, R(G), is the ring generated by the coefficients
of all irreducible unitary representations of G over the complex
field.
(5) For affine transformations of compact metric groups necessary and
sufficient conditions for ergodicity are known. The simplest case is
when G is a compact, connected, metric, abelian group. If Tx =
a>A(x) is an affine transformation of the compact, connected, metric,
abelian group G then the following are equivalent:
(a) T is ergodic.
(b) (i) Whenever Y°A = y f°r k > 0 then Y°A = Y, and,
(ii) the smallest closed subgroup containing a and BG (where
Bx = x-1-A(x) ) is G (i.e., [a,BG] = G).
(c) -3 xQ € G with {Tn(xQ): n 2 0} dense in G.
(d) m({x: {Tnx: n 2 0 is dense}}) = 1.
(Note that conditions (i) and (ii) reduce to the conditions given in
(3) and (4-) in the special cases. The equivalence of (a) and (b) was
investigated by Hahn, Hoare and Parry.)
Proof: First note that B is an endomorphism of G and
commutes with A.
(b) => (a). Suppose (i) and (ii) of (b) hold. If foT = f,
f € L2(m) let f = L biY-, Yi € G. Then
£ biYi(a)Y;L(Ax) = £ biYi(x). (*)
2
So if Yi, Y-°A, Y-°A , ... are all distinct then b, = 0 or else
YL |b.| < » is violated. Hence, if b. i 0 then Y.°An = Y- for
some n > 0, and by (i) YoA = Y. But then (*) implies Y.(a) = 1
and so Y^(x) =1 V x € [a,BG] and by (ii) Y. =1. So f is
constant a.e.
(a) => (d). This follows by the remark above.

28
(d) => (c) is trivial.
(c) => (b). It remains to show that if 3 x. € G with
{T xQ: n 2; 0} dense in G then conditions (i) and (ii) of (b) hold.
k ~ k
Suppose y°A = Y» k > 1, y € G. Let y-, = Y°B. Then YtCT x) =
k-1 k -1 k
Y-jCa-Aa- . . . -A a)Y (A x) = Y(a a a)Y.(x) = Y (x). Hence Y
assumes only the finite number of values Y..(xQ), ri(Txn)> •••>
Y-.CT xn) on the dense set {T x. : n £ 0} and hence assumes only
these values on G. Since G is connected y-, must be constant,
i.e., Yn = 1» Hence YA = Y and condition (i) holds.
If Ca,BG] ?! G 3 y * 1. Y « G, with Y(a) = 1, and Y(Bx) = 1.
Then Y(Tx) = y(*) and so y assumes only the value Y(xn) on the
dense set {T xn: n 2 0} and therefore y is a constant. Hence
Y s 1, a contradiction, and we have shown that condition (c) implies
(ii). //
When G is Kn the equivalence of (a) and (b) becomes: T =a-A
is ergodic iff
(i) the matrix [A] has no proper roots of unity (i.e.,
other than 1) as eigenvalues,
and (ii) [a,BKn] = Kn.
This is easily proved by a method similar to the one used in (M-)
for the endomorphism case.
Conditions for ergodicity of affine transformations of compact
nonabelian groups may also be found in Chu [1].
(6) The 2-sided (pQ,... ,p, _1)-shift is ergodic.
Proof: Let A = the algebra generated by finite unions of
measurable rectangles. Suppose T E = E, E € B. Let e > 0 be
given, and choose A € A 5 m(EAA) < e; thus
|m(E)-m(A)| = |m(EHA) + m(E\A) -m(AflE) - m(A\E) |
< m(E\A) + m(A\E) < e.

29
Choose n so large that B = T nA depends upon different coordinates
2
from A; so, m(B(lA) = m(B)m(A) = m(A) .
m(EAB) = m(T~nEAT~nA) = m(EAA) < e
and since EA(AflB) c EAA U EAB we have m(EA(AnB)) < 2e , hence
|m(E) - m(Afl B) | < 2e
and
|m(E)-m(E)2| £ |m(E) - m(A nB)| + |m(AnB) -m(E)2|
< 2E + |m(A)2 -m(E)2|
£ 2e + m(A)|m(A) -m(E)| +m(E)|m(A) -m(E)|
< te
2
since m(A),m(E) s 1. Since e is arbitrary m(E) = m(E) which
implies that m(E) =0 or 1. //
(7) By a similar argument, we see that the 1-sided (p.,... ,p, _,)-
shift is ergodic.
§5. The Ergodic Theorem
The first major result in ergodic theory was proved in 1931 by
G. D. Birkhoff [1].
Theorem 1.5: (Birkhoff Ergodic Theorem)
Suppose T: (X,B,m) -* (X,B,m) is measure-preserving (where we
allow (X,B,m) to be a-finite) and f €Lx(m). Then - £ f(T1(x))
n i=0
converges a.e. to a function f* € L (m). Also, f*oT = f* a.e., and
if m(X) < -, f f*dm = f fdm.

30
Note:
If T is ergodic then f = a constant a.e. and if m(X) < »
(X) J
,,,, , f dm a.e.
m(I
Motivation:
(i) Suppose T: (X,8,m) ■* (X,B,m) is measure-preserving and E € B.
For x € X, we could ask with what frequency do the elements of the
2
set {x, T(x), T (x), ...} lie in the set E?
Clearly T1(x) € E iff XET1(x) = 1, so the number of elements
, n-1 ,
of {x, T(x), . .., T (x)} in E is Z X T (x); and so the rela-
k=0 L
tive number of elements of {x, T(x), ..., Tn~ (x)} in E equals
- Y X TK(x). If m(x) = 1 and T is ergodic then - T_ X„T1(x) ->
ni=0 ' n i=0 L
m(E) a.e. by the note; and thus the orbit of almost every point of X
enters the set E with asymptotic relative frequency m(E).
(ii) We define the time mean of f to be
n-1
lim - E f(T1(x))
and the phase or space mean of f to be
f(x)dm.
-i-f
(i(X) J-
X
The ergodic theorem implies these means are equal if T is ergodic.
(The converse is also true.) So, it is important to verify ergodicity
for transformations arising in physics. This application to time
means and space means is more realistic in the case of a 1-parameter
flow {T.} of measure-preserving transformations. The ergodic theo-
1 fT 1
rem then asserts lim =- f(T x)dt exists a.e. for f € L (m) and
T-*~ l h r
equals m/v\ fdm i-n the ergodic case if the map (t,x) -* T.x -is

31
measurable.
An Application to Number Theory
Borel's Theorem on Normal Numbers :
Almost all numbers in [0,1) are normal to base 2, i.e.,
1_ , I the number of l's in the first n digits \ _^ 1_
n \ of the binary expansion of x € [0,1) J 2
a.e.
Proof: Let T: [0,1) -* [0,1) be defined by T(x) = 2x mod 1.
We know that T preserves Lebesgue measure and is ergodic, by
example 4- at the end of §4.
al a2 . •
Suppose x = — + —=- + ... has a unique binary expansion. Then
2 2
/al a2 a3 \ a2 a3
T(x) = T — +-i-+-4+...=-£• +-|+... . Let f (x) =
\ 2 2 2 / 2 2
Xr, ,n(x). Then
f1 iff a. ., = 1
' a. ,., a. ._ \ l+l
ftécx)) = f(îi+A + îiîi+ ...)
iff ai+1
Hence, the number of l's in the first n digits of the dyadic ex-
n-1
pansion of x is £ f(Tx(x)). Dividing both sides of this equality
i=0
by n and applying the ergodic theorem we see that
-, n-1 . /■ -,
- Z f(T x) ► Xr, n.dm = =■
n .t"n a.e. J [Jj,l) 2
(using the fact that the binary-rational points form a set of Lebesgue
measure zero). //
The ergodic theorem can be applied to give other number theoretic
results. Some are obtained in Billingsley [1] and Avez-Arnold [1],
We now consider some preliminaries to the proof of the ergodic
theorem.

32
Definition 1.3:
Let T: (X,8,m) -* (X,8,m) be measure-preserving. Define an
operator U_ on complex-valued functions on X by:
(UTf)(x) = f(T(x)).
We have UTLp(m) c L^(m) and, since T is measure-preserving
llU-fll = ||f|| . Let L£(m) denote the real-valued Lp(m) functions,
1 p p K
then UTL^(m) c L^(m).
To prove Birkhoff's theorem we need:
Theorem 1.6: (Maximal Ergodic Theorem)
Let U: LR(m) -* L_(m) be a positive linear operator (i.e.,
f£ 0 » Uf ; 0) which has norm £ 1. Let N > 0 be an integer.
Define fn = 0, f = f + Uf + U2f + ... + Un-1f, and F„ = max f ï 0.
0 n ' N Q£n£N n
Then fdm 2:
J{x:FM(x)>0}
0.
Proof: (due to A. Garsia) Clearly F.. € LR(m). We have for
0 < n £ N F.,ïf so, UFN 2: Uf by positivity, and hence
UFM + f > f .. . Therefore
N n+1
UFM(x) + f(x) > max f„(x)
N IsnsN n
= max f (x) when F.,(x) > 0
OsnsN n N
FN(x).
Thus f > FN - UFN on A = {x: F.,(x) > 0}, so

33
I f dm > [ F dm - [ UFMdm
- \x FNdm " |A UFNdm
- I FNdm " L UFN'
since F., = 0 on X\A.
N
dm since F., î 0 => UFN 2 0.
> 0 since IIUII < 1. //
Remark :
The conditions of Theorem 1.6 hold if U = U_ for measure-
preserving T.
Corollary 1.6 :
Let T: X -* X be measure-preserving. If g € LR(m) and
n n_1
ES = {x €X: sup ±. E g(Tm(x)) > a}
then
n>l n m=0
g dm > am(B (1A)
JB HA a
a
if T-1A = A and m(A) < -.
Proof: We first prove this result under the assumptions m(X) <»
and A = X. Let f = g -a, then B = U {x: F.,(x) > 0} so that
a N=0 N
fdm > 0 by Theorem 1.6 and therefore g dm > am(B ). In
Jb jb a
a a
the general case, using T|. in the place of T we see that
gdm î am(A PI B„) . //
a
'AflB a
Proof of Birkhoff's Theorem: It suffices to prove the theorem
]_ , n-1
for f € L_(m). Let f*(x) = lim - £ f(T1(x)) and f,(x) =
R n n i=0
n-1
lim è Y. f(T1(x)). We have f*oT = f*, f.°T = fA because if
n i=0

34
a„(x) = i- L fCT^-x) then (^) a„.. (x) - a(Tx) = ^^- . For real
n n .~_ V n ; n+1 n n
numbers j} < a, let
E„ B = {x€X: f.(x)<p, a<f*(x)}.
a ,p ft
Then T^E^p = E^ and
E^ n MX: sup ^ fCT^x)) > a} -- E^.
We now prove that m(E B)<" so that we can apply Corollary 1.6.
a j p
Suppose a > 0. Let C c E „ with m(C) < -. Then h = f - aX„
a ,p L
is integrable and by the maximal ergodic theorem
(f -aX )dm î 0. (HN defined analogously to the F.. in the
U {x:H'(x)>0}
N = 0
maximal ergodic theorem.) But C c 1J {x: HM(x)>0} so that
N=0 N
|f|dm ^ am(C). Therefore m(C) s — |f|dm for every subset
JX a Jx
of E Q with finite measure and hence m(E .)<». If a<0 then
a, p a,p
(3 < 0 so we can apply the above with -f and -p replacing f and
a to get m(E .) < ».
Πj p
n-1
Let B = {x € X: sup i £ fCT^O > a}. Then by Corollary 1.6:
a n>l i=0
f dm = fdm > am(E .(IB ) = am(E D) , i.e. ,
Je q Je 0ob a'P a a'P
a, p a, p a
[ fdm > am(E .) . (*)
Je a'P
If we replace f,a,p by -f,-p,-a respectively we get that
(-f)* = -f:i, (-f)ft = -f* and
I fdm s pm(E B). (**)
J -p aiP
o,p

35
So, if a > p then m(E a) = 0, and since
{x: fft(x)<f*(x)} ç (J Eajp
a, (3 rational
we have m{x: fft(x)<f*(x)} = 0 i.e., f*(x) = f (x) a.e. Therefore
n-1
-i 11 X .
— Yl f(T1(x)) converges a.e.
i=0
ft 1
To show f € L (m) we use the part of Fatou's Lemma that says
for non-negative iiytegrable functions g lim g dm < » implies
lim g is integrable. Let
gn(x)
i Z f(T1(x))
n i = 0
Then
| gndm =| ^1 f(T1(x)) dm < f |f
I dm
so that lim g dm < », and by Fatou's Lemma lim g = If I is
J n n ' ft '
integrable. Hence fft is integrable.
It remains to show that f dm = f*dm if m(X) < •». Let
D^ = {x €X: -<f*(x)<^i} where k € Z, nil. For each small
k n n '
e > 0 we have D, fl B , = D? and by Corollary 1.6
(H-e)
I fdm > (--e)m(DÎ1) so that
_ n k
fdm > - m(D^)
n k
(Aftft)
Then
[ f*dm £ *±Im(Dn) < I„()f) + f fdm (by (***))
J ^n '„n
Summing over k we get that

36
L
f*dm £ 2L12LL + | f dm V n > 1:
thus f*dm < fdm since m(X) < -. Applying this to -f instead
JX JX
of f gives (-f) dm £ -fdm i.e. , - fAdm - - fdm. Since
Jx Jx Jx * Jx
f = f* a.e. we get that f f*dm =:
* Jx '*
Hence, j f*dm = [ fdm. //
Corollaries 1.5:
(i) Let (X,B,m) be a probability space and T: X -* X measure-
preserving, then T is ergodic iff V A,B € B
i n_1
- Y. m(T XA n B) - m(A)m(B).
n i=0
Proof: (=>) Suppose T is ergodic. Putting f = X. in Theo-
1.5 gives i £ X.(Tx(x)) -> m(A) a.e. Multiplying by Xn:
n i=0 A B
, n-1
i- JT XA(T1(x))XB -> m(A)XB a.e.
By the dominated convergence theorem if we integrate we get
n-1
- 5~ m(T XA n B) -> m(A)m(B).
n i=0
(=) Let T E = E, E € B. Let A = B = E. Then - £ m(E)
n i = 0
2 2
m(E) so m(E) = m(E) , hence = 0 or 1. //
(ii) L^ Ergodic Theorem: (Von Neumann [1], [2])
Let 1 £ p < ». Let T be measure-preserving on the probability
space (X,B,m). If f € Lp(m), 3 f* € Lp(m) 5 f*oT = f* a.e.
n n-1
and || i Z fCT^-x) - f*(x) || - 0.

37
Proof: If g is bounded and measurable then g € Lp V p and
1 n_1 i
by the ergodic theorem we have that — £_ g(T x) -* g (x) a.e.
n i=0
Clearly g* € L°°(m) and hence g* € Lp(m). Also,
I — Z! g(T1x) - g*(x) | -* 0 a.e. and by the bounded convergence
n i=0 n
theorem, || - T g(T x) - g*(x)|| -+ 0 i.e., V e > 0 3 N(e,g) 3
n i=0 P
if n > N(e,g) and k > 0
i n_1 ,• n n+k-1
i Z g(Txx) - -4- I gcrSon <
n i=o n K i=o p
, n-l
Let f € Lp(m), and M (f)(x) = - T f(T1x). We must show that
n n ito
{M (f)} is a Cauchy sequence in Lp(mJ. Note that ||M (f)|| < ||f|| .
n n p p
Choose g € L°°(m) 3 ||f - gjj < eA; then
P
llMnf-Mn+kf"p £ "Mnf -"n^llp + »MnS " Mn+kS"p + "Mn+kS " Mn+kf «p
£ e/4- + e/2 + e/4- = e
if n > N(e/2,g) and k > 0. We have f*oT = f* a.e. because
(^)(M_inf)(x) - (M f)(Tx) - -(X-
n n+1 n n
§6. Mixing
We have seen that T is ergodic iff V A,B € B,
N-l
i £. m(T ^OB) -> m(A)m(B).
N i=0
Definitions 1. 4- :
(i) T is weak-mixing if V A,B € B
N-l
èr H |m(T XA PI B) - m(A)m(B) | -> 0.
N i=0

38
(ii) T is strong-mixing if V A,B € B
m(T"NA n B) -> m(A)m(B).
Note:
(i) T strong-mixing =» T weak-mixing.
(ii) T weak-mixing => T ergodic.
This is so because if {a } is a sequence of real numbers then
-. n-1 -, n-1
a -> 0 =» i Y |a.| -> 0 => - Y a. -+ 0.
n i=0 x n i=0 x
(Put a = m(T~nA flB) - m(A)m(B).)
n
(iii) An example of an ergodic T which is not weak-mixing is given
by T(z) = az on K, where a is not a root of unity. (See the
end of this section for the proof.)
(iv) There are examples of weak-mixing T which are not strong-
mixing. Kakutani has an example constructed by combinatorial methods,
and Maruyama constructed an example using Gaussian processes. Chacon
and Katok-Stepin also have examples. Indeed, if (X,B,m) is a
probability space, let t(X) denote the collection of all invertible
measure-preserving transformations of (X,B,m). If we topologize
t(X) with the "weak" topology (see Halmos [2]), the class of weak-
mixing transformations is of second category while the class of strong
mixing transformations is of first category.
The following result shows it suffices to check the convergence
properties on an algebra generating B.
Theorem 1.7 :
If T: X -* X is measure-preserving and A is an algebra
generating B then

39
(i) T is ergodic iff 'V A,B € A
n-1
- Z m(T-1AnB) ■* m(A)m(B),
n i=0
(ii) T is weak-mixing iff V A,B € À
, n-1
- £ |m(T-1AnB) - m(A)m(B)| -> 0, and
n i=0
(iii) T is strong-mixing iff V A,B € A
m(T~nAnB) -> m(A)m(B),
Proof: Let e > 0 and E,F € B. Choose En>Fn € A with
m(EAEQ) < e, m(FAFQ) < e. Then
and therefore
Hence
m((T nE nF)i(T_nEQ fl FQ)) < 2e
|m(T"nE HF) - m(T nEQ fl FQ) | < 2e.
n n_1 -v
=■ Z m(T E HF - m(E)m(F)
n k=0
|i £ Cm(T KE flF) - m(T"KEn f1Fn)]
'0 " '0'
- l£ jï. m(T-kE flF > - m(E0)m(F0)
n n_1
+ \~ T. m(E0)m(FQ) - m(EQ)m(F)|
n-1
+ |i I m(En)m(F) -m(E)m(F)|
n k= 0 °
n-1
- 2e + | ^ L m(T-kE0 fl FQ ) - m(E0)m(FQ) j + £ + e
for each n. (i) follows by the known behavior of the right hand side

to
of this inequality.
To prove (ii) first show
|m(T"kEnF) - m(E)m(F)| £ te + |m(T""kEQ fl FQ) -m(E0)m(FQ)| (*)
1 n_1
by leaving out " — H " from the above inequalities, and then take
n k=0
Cesaro averages of each side of (*).
(iii) follows immediately from the inequality (*). //
Theorem 1.8:
If {a } is a bounded sequence of real numbers then the
following are equivalent:
n-1
(1) i I |a£| - 0.
n i=0 1
(2) 3 J c Z , J of density zero, i.e.,
cardinality (J H{0,1,■■■ ,n-l})
n
) - »,
such that lim a = 0 provided n i J.
n n
(3) J Z Kl2 - 0.
n i = 0 1
Proof: If M c Z let aM(n) denote the cardinality of
{0,1,...,n-l} n M.
(1) = (2). Let Jk = {n € Z+: | an | > £•} (k > 0). Then
i=0
— j- a. (n). Therefore there exist integers 0 = ZQ < Z-. < Z„
J, c J c . . . . Each Jv has density zero since
such that for n t t, ,
1 , -, 1
— aT (n) < rpr .
n Jk+1 k+1
Set J = U [Jk+1 H C^v>^k+i^' We now show that J has density

m
zero. Since J c j c ..., if I < n < ^k+1 we have
j n [o,n)=[jn[o,y] u CJ n [£k,n)] c [jkn [o,-£k)] u [Jk+1 n [o,n)],
and therefore
i- aT(n) S kaT (.1.) + aT (n)] £ ka (n) + aT (n)] < i + -i_ .
n J n Jk k Jk+1 n Jk Jk+1 k k+1
Hence — aT(n) ■* 0 as n •*■ ~, i.e., J has density zero. If
n J
n > I. and n t J then n / J, ., and therefo
k k+1
re lan' < k+T* Hence
lim la I = 0.
t-» n '
J/n-*»
(2) => (1). Suppose |an| £ K V n. Let e > 0. There exists
N such that n î N , n i J imply |a | < e, and M such that
cij(n)
n î M implies < e. Then n 2 max(N ,M ) implies
£ n £ £
i n-1
- I |a.|
n iTo l!
I |aj + L |a.|
jn{0,l,.-. ,n-l} x if?jn{0,l,...,n-l} 1
< - aT(n) + e < (K+1)e.
n J
(1) «» (3). By the above it suffices to note that lim |a | = 0
Jjin-*»
iff lim |a | = 0. //
Jj>n— n
Corollary 1.8 :
T is weak-mixing iff V A,B € B 3 J(A,B) of density zero in
Z+ 5 lim m(T"nAnB) = m(A)m(B) iff V A,B € B
n/J(A,B)
n-1
i- Y. |m(T_:LA HB) - m(A)m(B)|2 -> 0.
i=0
Remark :
To say T is strong-mixing means that any set B € B as it

42
moves under T becomes, asymptotically, independent of a fixed set
A € B. T is weak mixing means B becomes independent of A if we
neglect a few instants of time. T is ergodic means B becomes
independent of A on the average.
The next result expresses the mixing concepts in functional form.
Recall that UT is defined on functions by Umf = f°T.
Theorem 1.9 :
Suppose (X,B,m) is a probability space and T: X -* X is
measure-preserving. Then
(a) T is ergodic iff for all f,g € L (m)
n n_1
± Z (U^f.g) -> (f,l)(l,g)
n i=0 T
iff for all f € L2(m)
i n"1
i Z (l£f,f) -> (f,l)(l,f).
n i = 0 i
2
(b) T is weak-mixing iff for all f,g € L (m)
i Z |(U™f,g> - (f,l)(l,g)
n i=0
iff for all f € L2(m)
- Z |(i£f,f) - (f,i)d,f)
n i=0
iff for all f € L2(m)
i- Z l(UJf.f) " (f,D(l,f)|2
n i=0 i
(c) (1) T is strong-mixing iff
(2) for all f,g € L2(m), (Ui£f,g) - (f,l)(l,g)
iff (3) for all f € L2(m), (U^f.f) - (f,l)(l,f),

43
Proof: (a), (b), and (c) are proved using similar methods. We
shall prove (c) to illustrate the ideas. Slight modification of this
proof will prove (a) and (b).
(2) => (1). This follows by putting f =XA> g=*B> for A,B € B.
(1) => (3). We easily get that for any A,B € B, (UTX.,XR) ->
(XA,1)(1,XR). Fixing B, we get that (u"h,XR) -> (h,l)(l,XR) for
any simple function h. Then, fixing h, we get that (U_h,h)
T A' B'
T..,/»B/ - vi.,J./vl,XB)
mh
(h,l)(l,h). So (3) is true for all simple functions.
o
Suppose f € L (m) , and let e > 0. Choose a simple function h
3 ||f-h|L < e, and choose N(e) so that n > N(e) implies
|(u£h,h) - (h,l)(l,h)| < e. Then if n>N(e)
.|(u!Jf,f) - (f,l)(l,f)| £ | (u£f,f) - (u£h,f)|
+ |(u£h,f) - (u£h,h)| + |(u£h,h) - (h,l)(l,h)|
+ |(h,l)(l,h) - (f,l)(l,h)| + |(f,l)(l,h) - (f,l)(l,f)|
£ |(UT(f-h),f)| + |(UTh,f-h)|
+ e + | (l,h)| | (h-f,l) I + |(f,l)| |(l,h-f) |
< ||f-h||2||f||2 + ||f-h||2||h||2 + e + ||h||2||f-h||2 + ||f||2||h-f||2
by the Schwartz inequality
£ e||f||2 + e(||f|l2 +e) + e + (||f||2 +e)e + e||f||2.
Therefore (UTf,f) -> (f,l)(l,f) as n -> -.
(3) => (2). Let f € L2(m) and let Hf denote the smallest
2
(closed) subspace of L (m) containing f and the constant functions
and
satisfying UTHf c Hf.
Ff = {g € L2(m): (U^f,g) -> (f,l)(l,g)}
2
is a subspace of L (m), is closed, contains f and the constant

44
functi ons and is U™ invariant so i"t contains H^ • If g x H^ "then
(U^f.g) = 0 for n î 0 and (l,g) = 0 and therefore H^ c Ff.
Hence Ff = L2(m). //
Definition 1.5:
If T: X -* X is measure-preserving, define TxT: XxX -* XxX by
(TxT)(x,y) = (T(x),T(y)). This is a measure-preserving transformation
on (XxX,8x8,mxm) by Theorem 1.1 since it is measure-preserving on
measurable rectangles and hence on finite disjoint unions of such
rectangles.
Theorem 1.10 :
If T is a measure-preserving transformation on a probability
space X then the following are equivalent:
(1) T is weak-mixing.
(2) TxT is ergodic.
(3) TxT is weak-mixing.
Proof: (1) => (3). Let A,B € B, C,D € B. 3 J±^2 of density
zero such that
lim m(T"nAriB) = m(A)m(B)
n-*»
lim m(T-ncriD) = m(C)m(D).
n(J2
n-*»
Then
lim (mxm){(TxT) n(AxC) fl ( BxD) } = lim m(T_nA n B)m(T nC n D)
n^JjUJ2 n<JjUJ2
= m(A)m(B)m(C)m(D)
= (mxm)(AxC)(mxm)(CxD).
Thus the proper relationship holds for rectangles and hence for finite
disjoint unions of these rectangles. These we know form an algebra F

45
which generates the a-algebra B. By Corollary 1.8 we have
, n-1
- Z |m(T-1AnB) - m(A)m(B)| -* 0 V A,B € F and the result follows
n i=0
by Theorem 1.7.
(3) ■=> (2) is clear.
(2) => (1). Let A,B € B. We have that
, n-1 _. , n-1
- Z m(T 1AnB) = - Y. (mxm)((TxT)-:L(AxX) H (BxX))
n i=0 n i=0
-* (mxm)(AxX)(mxm)(BxX) by (2)
= m(A)m(B).
Also
-, n-1 . „ . n-1
£ Z (m(T-1A flB)r = - E (mxm)((TxT)-:L(AxA) n (BxB))
n i=0 n iTo
(mxm)(AxA)(mxm)(BxB) by (2)
= m(A)2m(B)2.
Thus
-, n-1 . „
=- Z {m(T_:LA flB) -m(A)m(B)r
n i=0
k Z {m(T-1AnB)2 -2m(T XA nB)m(A)m(B) +m(A)2m(B)2}
n i=0
2m(A)2m(B)2 - 2m(A)2m(B)2
Therefore T is weak-mixing by Corollary 1.8. //
Definition 1.6:
Let T: (X,B,m) -* (X,B,m) be a measure-preserving transformation
on a probability space. We say that X is an eigenvalue of T,
(X € C) if 3 f i 0 € L2(m) 3 UTf = Xf in L2(m); i.e.,
f(Tx) = Xf(x) a.e. We call f an eigenfunction corresponding to X.

46
Remarks :
(i) If X is an eigenvalue of T then |\| =1 since
||f||2 = ||uTf||2 = (uTf,uTf) = (\f,\f) = |x|2||f||2.
(ii) X = 1 is always an eigenvalue corresponding to any constant
eigenfunction.
Definition 1.7 :
We say that T: X -* X has continuous spectrum if 1 is the only
eigenvalue of T and the only eigenfunctions are the constants.
Observe that T has continuous spectrum iff X = 1 is the only
eigenvalue and T is ergodic.
We shall need the following result from spectral theory to prove
the next theorem. The proof can be found in Halmos [3].
Spectral Theorem for Unitary Operators :
Suppose U is a unitary operator on a complex Hilbert space H.
Then V f € H, 3 a unique finite Borel measure \if on K 5
(Unf,f) = [ Xnd|^(X) V n € Z.
JK r
If T is an invertible measure-preserving transformation then
Um is unitary, and if T has continuous spectrum then M.f has no
atoms for all f € L (m) with (f,l) = 0.
Theorem 1.11:
If T is an invertible measure-preserving transformation of a
probability space then T is weak-mixing iff T has continuous
spectrum.
Proof: (=>). Suppose fT = Xf a.e., f € L2(m). If X ?! 1
then integration gives (f,l) = 0 and by the weak-mixing property

1*7
, n-1
i I |(l£f,f)| - 0
i.e. ,
n i = 0
i n-1
n i=0
Since |X1| = 1, this gives (f,f) = 0 and therefore f = 0 a.e.
If X = 1 then f = constant a.e. by the ergodicity of T.
(=). Suppose T has continuous spectrum. We show that if
f € L2(m) then
, n-1 . „
£ I! |(U^f,f) - (f,l)(l.f)| - 0.
n i=0 i
If f is constant a.e. this is true. Hence, all we need to show is
that (f,l) = 0 implies
k% i«4f,f)i2 - o.
By the spectral theorem it suffices to show that if p._ is a
continuous (non-atomic) measure on K then
n-1
£ f if ^dMX)!2 - 0.
n i=0 J r
We have
n-1 r on n-1
\ L if xVfU)|2 = i. £ ^J X^^CX) • J X ^(X))
= ^ Zo(f xid"fU) ■ I T"id^f(T))
= J7 .Z (XT)1d(|j.fxM.f.)(X,T) (by Fubini's Theorem)
1-0 KxK
\\ ( H- .? C^)1) d(nfxnf)(X,T).
KxK

48
If (\,t) is not in the diagonal of KxK then
-> 0
as n -* ». Since |j.f has no atoms ( |j.fX|j.f) (diagonal) = 0 and
therefore the integrand ■* 0 a.e. The integrand has modulus £ 1, so that
we can apply the bounded convergence theorem to obtain the result. //
We now investigate the mixing properties of the examples
mentioned in §1.
Examples :
(1) I = identity on (X,B,m). I is ergodic iff all the elements
of B have measure 0 or 1 iff I is strong-mixing.
(2) T(z) = az on K. T is never weak-mixing since if f(z) = z
then f(Tz) = f(az) = af(z) and f t constant. (This has used the
trivial part of Theorem 1.11.)
(3) No rotation on a compact group is weak-mixing. We have already
mentioned that if T is ergodic then the group G is abelian; and
then if Tx = ax and y is any character of G we have y(Tx) =
Y(a)y(x), which shows that T does not have continuous spectrum.
(M-) Endomorphisms of compact metric groups are strong-mixing iff
weak-mixing iff ergodic.
Proof : We shall give the proof when G is abelian. It
suffices to show that if the endomorphism A: G -* G is ergodic then A
n
is strong-mixing. If y,6 * G then (U-y,6) = 0 eventually unless
n
Y = 6 = 1. So always (U y,6) -* (y,l)(l,6). Fix 6 € G. The
collection
H5 = {f € L2(m): (U^f,6) -> (f ,1) (1,6 ) }
2
is a closed subspace of L (m). (To check H is closed, suppose
t n-1
n i = 0 n
1 - (Xt)1
1 -(Xt)

49
f, € H and f, -* f € L2(m). For 6=1 it is clear that H, =
k k 6
L2(m). So suppose (1,6) = 0. Then
|(U^f,6)| < |(u£f,6) - (U^fk>6)| + |(U^fk>6)|
S llf-fkll2l|6|l2 + |(UAfk,6)| (by the Schwarz inequality)
|f-fkll2+ KUAfk'6)| < e
if n 2 N(e) where k is chosen so that ||f-fkH2 < e/2 and N(e)
is chosen so that n > N(e) implies |(U?f.,6)| < e/2.) Since H
^ 2 2
contains G, it is equal to L (m). Fix f € L (m) and consider
2 n
Lf = {g € L (m) : (UAf,g) -* (f,l)(l,g)}. Lf is a closed subspace of
2 * • 2
L (m), contains G by the above, and so equals L (m)• Hence A is
strong-mixing. //
(5) An affine transformation T = a-A on a compact metric abelian
group is strong-mixing iff it is weak-mixing iff A is ergodic.
Proof: We shall give the proof in the case when G is connected.
Let Bx = x~ A(x) and recall that T is ergodic iff
(i) y°A = y k > 0 implies y°A = y, a^d
(ii) [a,BG] = G.
If A is ergodic then BG = G since the endomorphism B of G
is one-to-one. Choose b € G so that B(b) = a. Put 4>(x) = bx;
then 0T = A0. ^ preserves Haar measure m on G and hence induces
2
a unitary operator U on L (m). We then have that U„ and U.
are isomorphic as Hilbert space operators by the induced conjugacy
vu* -- vv
Now A is strong-mixing by (M-), and hence U. satisfies the
conditions of part (c) of Theorem 1.9. But then UT satisfies these
conditions, and hence T is strong-mixing.

50
Conversely if T is strong-mixing and A is not ergodic then
by (i) y°a = Y for some y i 1. But then
|(U?Y.Y)| = |(Y(a)Y(Aa)...Y(An"1a)Y,Y)| = HyIIj = X
which does not converge to |(y,D(1,y)| = 0> contradicting the
mixing of T. So if T is strong-mixing then A is ergodic. //
(6) The 2-sided (p.,...,p, _.)-shift is strong-mixing. This is proven
by doing the easy verification on measurable rectangles, then on their
disjoint finite unions, and then applying Theorem 1.7.
(7) Similarly, the 1-sided (pn,...,p, _.)-shift is strong-mixing.

Chapter 2 : Isomorphism and Spectral Invariants
§ 1. Isomorphism of Measure-Preserving Transformations
What should we mean by saying that two measure-preserving
transformations are the "same"? We must bear in mind that sets of measure
0 do not matter from the point of view of measure theory.
Examples :
2
(1) Let T be the transformation Tz = z on the unit circle K
with Borel sets and Haar measure, and let S be given by Sx = 2x
mod 1 on [0,1) with Borel sets and Lebesgue measure. Consider the
map <j> : [0,1) ■* K defined by x-*enx. 4> is a bijection and
preserves measure (check on finite unions of intervals and use Theorem
1.1). Also <t>S = T*. So, we want to regard T and S as the "same".
(2) Again, let S be the transformation Sx = 2x mod 1 on [0,1)
with Borel sets and Lebesgue measure, and let T„: X -* X be the
1-sided (Jj,Jj)-shift. Define \|r : X -* [0,1) by
a a„ a
^(a1,a2,a3,...) = _ + -^ + _ + . . . .
\|r is not one-to-one only at points (a. ,a.,...) whose coordinates are
constant eventually. \|r, though, is onto and \|rT„ = S\|r. Also \|r
preserves measure; we can check this out on dyadic intervals and then on
their finite disjoint unions and apply Theorem 1.1.
Suppose D is the set of points of the space X of the 1-sided
(Js,Js)-shift which have constant coordinates eventually. Then T„ D = D
and so T~ (X\D) = X\D. Let D„ consist of the dyadic rationals in
[0,1). Then S-1D2 = D2, so that S-1([0,1)\D2) = [0,1)\D2.

52
We see that the diagram
T
X\D - > X\D
\|r I I \|c 1:1 onto
[0,1)\D2 —^—> [0,D\D2
commutes.
We would like to consider these transformations as isomorphic
since, after removing sets of measure zero, we can throw one to the
other by an invertible measure-preserving transformation.
Definition 2.1:
Suppose (X,,B..,m,) and (X2,B2,m„) are probability spaces
together with measure-preserving transformations T, : X, -* X,,
T„: X„ -* X„. We say that T, is isomorphic to T„ if 3 M, € B ,
m1(M1) =1, M2 € B2, m2(M2) = 1 3
(i) T1M1 ^ M]_, T2M2 i M2, and
(ii) 3 an invertible measure-preserving transformation
4>: M]_ -> M2 3 *T1(x) = T2«(x) V x € M]_.
We write T, = T2. (In (ii) the set Mi (i=l,2) is assumed to be
equipped with the a-algebra Mi fl Bi = {M. fl B | B € B^ and the
restriction of the measure m. to this a-algebra.)
Remarks :
(a) ^ is an equivalence relation.
(b) T± ^ T2 =• Tn ^ Tn, V n > 0.
(c) If T, and T„ are invertible we can take M1,M2 so that
k k
1M1 = Ml' T2M2 = M2' we 3ust take H TiMi' H T2M2 as the new
T
sets .

53
§ 2. Conjugacy of Measure-Preserving Transformations
Although the notion of isomorphism, introduced above, is useful
in practice the following is mathematically more natural.
Given (X,B,m) we define an equivalence relation on B by
saying that A ^ B iff m(AAB) = 0. Let B denote the set of
equivalence classes. g is a Boolean a-algebra under the operations
induced from the usual operations on B. m induces a measure m
on 8. We call (B,m) a measure algebra. Note that for B € 8 , X„
2
is a uniquely defined member of L (m).
Suppose T: X -* X is measure-preserving. If A ^ B then
T-1A ^ T~ B; so we have a map T~ : B -* B which is defined by
T~ (?) = î^_B- T. preserves unions, intersections, and complements,
and m(T-1B) = m(g).
Definition 2.2:
A map * : (B„,m„) -* (8. ,51.) of measure algebras is called an
isomorphism of measure algebras if * is a surjective bijection and
preserves complements and countable unions and
n^CSCBj)) = m2(g2) V §2 € g2.
Definition 2.3:
We say that T±: X± ■* X±, T2 : X2 -♦ X2 are conjugate if 3
measure algebra isomorphism * : (B„,m„) ->■ (B,,^) such that
-1 -1
*T2 = T1 *.
Remarks :
(1) Conjugacy is an equivalence relation.
(2) T1 ^ T2 =» T1 and T2 are conjugate.
Just let * = <t> , which is uniquely defined although <t> is
not defined on the whole of X,.

54
In Lebesgue spaces (i.e., probability spaces isomorphic to a sub-
interval of [0,1] with Lebesgue measure possibly together with
countably many atoms) conjugacy implies isomorphism. In particular,
a compact separable metric space with a completed Borel measure is a
Lebesgue space.
§3. Spectral Isomorphism
Suppose T: X ■* X is a measure-preserving transformation on a
2 2
probability space (X,B,m). We have defined U_: L (m) -* L (m) by
f h- foT, and noted that V f,g € L2(m) we have (U f,U„g) = (f,g).
Also, if T is one-to-one, U_ is unitary. A spectral property
of T is a property of U_.
Definition 2.4:
Measure-preserving transformations T, on (X-,8.,m1), and T„
on (X„,B„,m„) are spectrally isomorphic if 3 a linear operator
2 2
W: L (m2) -* L (m ) such that
(i) W is invertible
(ii) (Wf,Wg) = (f,g) V f,g € L2(m2)
(iii) UT W = WUT .
(The conditions (i), (ii) just say that W is an isomorphism of Hil-
bert spaces.)
Remarks :
(1) Spectral isomorphism is an equivalence relation.
(2) If *: (8.,m.) -* (B.jm,) is a measure algebra isomorphism then
* induces an invertible linear maj
VX„ = X.,Rv, with the properties:
2 2
* induces an invertible linear map V: L (m„) -* L (m.), by
(a) (Vf,Vg) = (f,g) V f,g € L2(m2).

55
(b) V, V- map bounded functions to bounded functions.
(c) V is multiplicative on bounded functions.
Proof: V is defined as follows. Let B„ ç B2> then
V(x„ ) = x /n > which is unambiguous in L (m„). We then extend V
2
to simple functions and then to L (m„) functions. The properties
of $ guarantee this can be done. (a), (b), and (c) are proved by
checking first for characteristic functions, then for simple functions,
2
and then extending to the whole of L (m„). //
(3) If T, and T„ are conjugate then they are spectrally
isomorphic.
Proof: Suppose $ : (B2>m2^ "* C B -■ » Ji-, ) is an isomorphism of
measure algebras such that $T~ = T, *. Let V be defined as in
remark (2). It remains to check that
VUT = UT V.
2 ''I
First, on characteristic functions
UT V(X ) = UT (X.R ) = X _ = X _ = V(X ) = VU„ (X ).
1 ~2 1 ~2 Ti *?2 *~2 ~2 ~2 ~2 2 2
Therefore UT V and VU_ agree on characteristic functions and
1 i2
hence on linear combinations of characteristic functions. By their
continuity we have UT V = VU_ . //
il i2
The following tells us when spectral isomorphism implies conjugacy.
Theorem 2.1:
2 2
An invertible isometry V: L (m„) -* L (m ) is induced by an
isomorphism of measure algebras (in the same sense of remark (2)) if both
V and V take bounded functions to bounded functions and V(fg) =
2
V(f)V(g) whenever f and g are bounded and in L (m„).

56
Proof: Let B„ € B„ . We have Xn = Xn so that
V(Xn ) = V(XR )V(Xn ) = V(XR ),
B2 B2 B2 B2
and we see that V(XD ) takes 1 and 0 as its only values. Thus
B2
3 B.. € B.. such that V(Xn ) = Xn a.e. We define $: (B„,m„) -*
1 1 D« D, ~Z ~Z
C B- ,m, ) by *(B_) = B,. This is unambiguous since if i,(B JAJ = 0
then ||x„ - X. || = 0 so that ||V(xn ) - V(x. ) || = 0. Clearly, V
B2 A2 2 A2
is induced by * in the sense of remark 2.
We now show that $ is an isomorphism of measure algebras.
First, $ is invertible by doing the above for V . Also,
m2(B2) = m2(B2) = J X^ dm2 = (X^.x^)
= (VX§2,VXR2) = (Xi(§2),Xi(|2)> = mi(*B2).
It remains to show that $ preserves complements and countable
unions. First note that since V is norm-preserving and maps
characteristic functions to characteristic functions, V(l) = 1.
2
Since Xn + Xv . D =1 in L (m„) applying V to both sides
~2 wo^S?
gives X$B + X$(x B j = 1 so X^Sj^ = *(X2\B2). Therefore $
preserves complements.
Suppose B,C € B2. Then
XBUC XB + XC XBflC XB + XC XBXC"
Taking V of both sides we get:
X*(BUC) = X$(B) + X*(C) X*(B)X$(C) = X$(B)U*(C)'
Thus $(B UÇ) = *(§) U*(C) and hence, (by induction) preserves all

57
finite unions.
Let B1,B2,...,Bn,... € B2, then
X -* X a.e.
Û B£ Û gi
i=l i=l 1
2
and also in L (m„) by the bounded convergence theorem. Since V is
an isometry it is continuous, so,
v/x \ -> v/x \ = X .in L2(m1).
i=l
On the other hand,
iW \ 9w <9^)
v/x \ = x = x
2
by the above and so converges to X in L (m.). Therefore
Û *§•
i=l 1
*( U §i) = U *B£. //
i=l i=l
Corollary 2.1:
If T, : X, ->■ X, , T„: X„ ->■ X„ are measure-preserving and if
U V = VU for V: L2(m ) -♦ L (m ) satisfying the conditions of
il i2 l x
Theorem 2.1, then T, and T„ are conjugate.
§4. Spectral Invariants
Definition 2.5:
A property P of measure-preserving transformations is a
isomorphism
conjugacy invariant if the following holds:
spectral

58
Given T1 has P and T2 is
isomorphic
conjugate to T,,
_spectrally isomorphic
then T„ has property P.
Note:
A spectral invariant is a conjugacy invariant, and a conjugacy
invariant is an isomorphism invariant.
Theorem 2.2 :
The following are spectral invariants of measure-preserving
transformations :
(i) Ergodicity
(ii) Weak-mixing
(iii) Strong-mixing.
Proof; (i) T is ergodic iff {f € L2(m): UTf = f} is a one-
dimensional subspace.
(ii) T is weak-mixing iff 1 is the only eigenvalue and T
is ergodic.
(iii) Suppose WUT = U_ W and T. is strong-mixing. We have
i2 il x
to show that
(U h,k) -+ (h,l)(l,k) V h,k € L2(m ).
i2 L
This is true if h is constant or if k is constant, so assume
(h,l) = 0 = (k,l). Since 1-^ is ergodic then T„ is ergodic by (i)
and since W sends the invariant functions for T„ onto those for
2
T-,, W maps the subspace of constants in L (m„) onto the subspace
of constants in L (n^). So (Wh,l) = 0 = (l,Wk). Since W
preserves the inner product,
(UT h,k) = (WU h,Wk) = (UT Wh,Wk) ■*■ 0
i2 L2 il
since T, is strong-mixing. Therefore T„ is strong-mixing. //

59
§5. Examples
Recall that T, is isomorphic to T„
=» T, is conjugate to T„
=> T, is spectrally isomorphic to T„
and the converse of the first implication holds in all "decent"
measure spaces.
(1) Consider T1>T2: K -* K given by T-^z) = a^, T2(z) = a2z
where a, is a root of unity and a„ is not a root of unity. T. is
not ergodic while T„ is ergodic. Hence they cannot be spectrally
isomorphic.
(2) Let T(z) = az where a is not a root of unity. We know that
2 2
T is ergodic but not weak-mixing. Consider A: T -* T defined by
A(z,w) = (zw,z). Since none of the eigenvalues of the matrix I 1
are roots of unity, A is weak-mixing. Hence T and A are not
spectrally isomorphic.
(3) Let at least two of the numbers {p-, >P2 > • • • >P } be non-zero, and
n
^T p. = 1. Let the same be true for the numbers {q,,...,q }. We
i=l 1 i m
claim that the 2-sided (p,,...,p )-shift and the 2-sided (□.,...,q )-
1 n ^1 ^m
shift are spectrally isomorphic but not necessarily conjugate. A
consideration of entropy shows that they need not be conjugate. (See
Chapter 4.)
Consider the special case of the (Js,Js)-shift T, with
X = TT {-1,1}. A basis for L ({-1,1}) consists of the constant
function 1 and the identity map sending
(-1) h- (-1), 1 h- 1.
2 -t-r- 2
Moreover, L (TTX.) is the tensor product of the spaces L (X.) so
2
that we have an orthonormal basis for L (X) consisting of all

60
functions X -* C of the form:
g0({xn}) - 1
and, for n, < n„ < ... < n
g ( {x }) = x -x •. . . -x
6n1>...,nr n1 n± n2 np
Note that
Vn1,...,nr^xn>) = ^n,,. . . ,nr°T> < <xn»
= xn1+l'xn2+l'-"'xnr+l = gni+l ,n2+l,. . . ,nr+l( {xn}) '
that is,
UTgn1,...,nr = gn1+l,...,nr+l*
2
So we have an orthonormal basis for L (X) of the form;
f0 5 i, {u;fl}n€Z , {u;f2>n€Z , ... .
Diagramatically, the basis has the form
f0 5 X
.., u~ f1, u~ f1, f1, uTf1, uTf19
• • • , Um o' T 2' 2' ^t,-^2' Urn^2'
Definition 2.6:
An invertible measure-preserving transformation T: X ■* X h
countable Lebesgue spectrum if there exists an orthonormal basis
L2(X) of the form:
fQ = 1, {U^fj} j 2 1. n € Z,
i.e., a basis as in (*) above.

61
Remarks :
(1) Any two transformations with countable Lebesgue spectrum are
spectrally isomorphic.
Proof : If T.-X-X, S: Y -* Y have bases:
n n€Z 2
f0 s 1, {UTfj>J€Z+ for L (X)
n n€Z 9
g0 s 1' <USgj>J€Z+ for L (Y)
we define W: L2(Y) -> L2(X) by gQ -> fQ, Ugg. -> UTf. and extend by
linearity. Thus WU<, = UTW and S and T are spectrally isomorphic.
(2) Using a similar method to the one used above for the (Jj ,Jj)-shift
one can show that if at least two of {p,,p„,...,Pn> are non-zero
then the 2-sided {p, ,p„ ,...,p }-shift has countable Lebesgue spectrum.
Theorem 2.3:
If T has countable Lebesgue spectrum it is strong-mixing.
Proof: Let {f ,U„f : n€Z, m>0} be the basis. Then,
clearly, as p -* »
(UT°UTfm'UTfq) - (UTfm5l)(1'UTfq} V k'n € Z' m'q " °'
Fix k and q and consider
Hk,q = <f € L2(m): (U^f,U^fq) -> (f,1)(l.U^f )}.
2
tf, is a closed subspace of L (m) (c.f. proof in example (4) §6
k ,q
Ch. 1) and contains the basis {fQ,UTf : n €Z, m>0} and hence is
equal to L2(m). Fix f € L2(m) and let Lf = {g € L2(m):
P 2
(Urpf,g) -» (f,l)(l,g)}. Lf is a closed subspace of L (m) , contains
the basis by the above, and therefore is equal to the entirety of
2
L (m). Hence
(U^f,g) -> (f,l)(l,g) V f,g € L2(m). //

62
Suppose A: G -* G is an ergodic automorphism of a compact
abelian metric group. Then the automorphism A: G -* G has no finite
orbits except for the orbit of the identity. (This is what the er-
2
godicity of A says.) Since the characters form a basis for L (m)
we can conclude that A has countable Lebesgue spectrum if we can
show there are infinitely many distinct orbits of A. This is proved
in Halmos [2].
In Chapter 4 we shall consider a whole class of transformations
with countable Lebesgue spectrum.

Chapter 3 : Measure-Preserving Transformations
with Pure Point Spectrum
In this chapter we study a class of measure-preserving
transformations for which the conjugacy problem is solved and for which
spectral isomorphism implies conjugacy.
§1. Eigenfunctions
Suppose T is an ergodic measure-preserving transformation of a
probability space (X,B,m). Suppose \ is an eigenvalue correspond-
2
ing to the eigenfunction f, i.e., f t 0, f € L (m), U_f =
\f € L2(m) ( (foT)(x) = \f(x) a.e.). Then
(1) |\| = 1 and |f| is a constant a.e.
We have f(T(x))f(T(x)) = \\f(x)f(x) a.e. Integrating both
sides we get that ||f||2 = |\|2||f||2. Therefore |\| = 1. Also
|f(T(x))| = |\||f(x)j a.e. = |f(x)| a.e. Thus |f| is a T-invariant
function and, since T is ergodic, |f| = a constant a.e.
(2) Eigenfunctions corresponding to different eigenvalues are
orthogonal.
Suppose \ i |j., U_f = Xf, UTg = |j.g. Then
(f,g) = (UTf,UTg) = (Xf,^g) = \il(f,g)
and X|j. i 1 implies (f,g) = 0.
(3) If f°T = Xf, goT = Xg then f = c-g where c is some
constant.
By (1) g i 0, so (f/g)°T = f/g which must be constant since
T is ergodic.
So (2) and (3) show that eigenspaces are 1-dimensional and

64
mutually orthogonal.
(4) The eigenvalues of T form a subgroup of K.
If foT = Xf, g°T = M-g then (fg)°T = Xjlfg.
2
By (2) if L (m) is separable then the group of eigenvalues is
countable.
§2. Pure Point Spectrum
Definition 3.1:
An ergodic measure-preserving transformation T on a
probability space (X,B,m) has pure point spectrum (discrete spectrum) if
2
there exists an orthomormal basis for L (m) which consists of eigen-
functions of T.
The following theorem shows that the eigenvalues determine
completely whether two such transformations are conjugate or not.
Theorem 3.1: (Discrete Spectrum Theorem - Halmos and Von Neumann
[1], 1942)
The following are equivalent for ergodic measure-preserving
transformations T, and T„ each having pure point spectrum:
(1) T. and T„ are spectrally isomorphic.
(2) T, and T„ have the same eigenvalues.
(3) T, and T„ are conjugate.
Proof: (1) =» (2) is trivial.
(3) => (1) is always true (see §3 of Chapter 2).
2
(2) => (1). For each eigenvalue X, choose f, € L (nu.) ,
2
g, € L (m„) such that
and
\fX - XfX> UT/X = XS>
IM "" l*xl = x-

65
2 2
We define W: L (m„) -* L (m ) by W(g,) = f. and extending by
linearity. We readily see that W is a bijective isometry; moreover
WUT = UT W by checking this on the g. .
i2 il X
(2) => (3). To prove this we need the following result:
Theorem 3.2 :
Let H be a discrete abelian group and K a divisible subgroup
of H (i.e., V k € K and Vn>0 3 a € K 3 an=k). Then
there exists a homomorphism ^ : H -* K such that <t>\K = identity
(i.e., K is an algebraic retract of H).
Proof: Let R consist of all retracts onto K from supergroups
of K in H, i.e., R consists of all pairs (M,*) where H>M>K
and <t> : M -* K is a homomorphism such that <j> | „ = identity. R is
non-empty as (K,idK) € R. We order R by extension, i.e.,
(M1,«1) < (M2,«2) if M1 s M2 and *2Im = *i* This is a partial
ordering and every linearly ordered subset has an upper bound. So,
by Zorn's Lemma there exists a maximal element, say (L,p), of R.
We claim that L = H. Suppose not, then consider g € H\L and
let M be the group generated by g and L.
Case 1: If no power of g lies in L then every element of M can
be uniquely written in the form g a where a € L, i € Z. We define
\|r: M -* K by \|r(g a) = p(a). We can easily check that \|r is a
homomorphism and that \|r | „ = id,,. This then contradicts the maximality of
(L,p).
Case 2: Let n be the least positive integer such that g € L.
Each element of M can be uniquely written as g a, where a € L,
0 s i S n-1. Since K is divisible, let g. € K be such that
p(g ) = gn« Then \|r(g a) = g.p(a) defines a homomorphism of M
into K such that \|r | „ = id, . Again, we have contradicted the
maximality of (L,p).

66
Thus it follows that L = H. //
We now prove that (2) => (3). Let A denote the group of
eigenvalues of T, = the group of eigenvalues of T„. Fix X € A. Let
2
f. € L (nu) be chosen so that |f. | = 1, UT f. = \f. and observe
2 2
that {f\: X € A} is a basis for L (m,). Also, choose g, € L (m„)
so that |g.| = 1, U_ g. = Xg and observe that {gx: ^ € A} is a
2
basis for L (m„).
UT.fX^ = X^ V \„x € A
l
and also
UT (f,-f ) = f,(T)-f (T) = (X|^)(f.-f ).
By (3) of §1 there exists a constant r(X,|j.) € K such that
f,(x)f (x) = r(X,u)f, (x) a.e.
X |j. X|j.
Let H denote the collection of all functions X •*■ K. Clearly,
H is an abelian group under pointwise multiplication. Moreover, K
is a subgroup of H if we identify constant functions with their
values.
By the previous Theorem 3.2 there exists a homomorphism ^: H -* K
such that, *IK = idK. Let f* = $(fx)fx; then |f*| = 1, UTf* = \f*
and {f?: X € A} is a basis for L (m.). Also,
f*f* = *(f,)«(f )f.f = 0(f.f )f,f
0(r(\,M.))*(fx )fXM_r(\,M.)
r(\,M.)*(fx )r(\,M.)fx
Thu
s for all intents and purposes we can assume that ^\^u ~ ^\u an<^
gXg|^ = gX|T

67
2 2
Define W: L (m„) -* L (m.) by W(g)) = f\ and extend by
linearity. W is bijective, linear and preserves the inner product.
Also, WU_ = UT W. If we can show that W is multiplicative then
i2 il
W is necessarily induced by an isomorphism of measure algebras (by
Corollary 2.1) and hence T and T„ are conjugate. But,
W(g,g ) = W(g. ) = f, = f,f = W(g,)W(g ).
2
Let h,k be bounded functions in L (m„). If we fix g and let a
' 2
finite linear combination of g s converge to h in L (m„) we ob-
tain that W(hg ) = W(h)W(g ). Then if we let a finite linear combi-
nation of g 's converge to k in L (m„) we get that W(hk) =
W(h)W(k). It follows from this that W maps bounded functions to
bounded functions since this is also true for bounded h and any
2 2
k € L (m„), and then W(h) is bounded since W(h)f € L (nu) for
all f in L2(m1). //
Corollary 3.3:
If T is an invertible ergodic transformation with pure point
spectrum then T and T are conjugate.
Proof: They have the same eigenvalues. //
§3. Group Rotations
Example •■
Let T: K -* K be defined by T(z) = az where a is not a root
of unity. We know that T is ergodic. Let f : K -* C be defined
n
by f (z) = z where n € Z.
n
f (Tz) = f (az) = anzn = anf (z).
n n n
Thus f is an eigenfunction with eigenvalue a . Since the i^-n)
2
form a basis for L (K) we see that T is ergodic and has pure

68
point spectrum.
These ideas carry over to rotations on compact abelian groups.
Theorem 3.3:
Let T, (T(g) = ag) be an ergodic rotation of a compact abelian
group G. Then T has pure point spectrum. The eigenfunctions of T
all consist of constant multiples of characters, and the eigenvalues
are {y(a): y € G}.
Proof: Let y € G. Then
Y(Tg) = r(ag) = r(a)Y(g).
Therefore each character is an eigenfunction and so T has pure point
spectrum. If there is another eigenvalue besides the members of
{y(a): y € G} then the corresponding eigenfunction would be
orthogonal to all members of G, by CO of §1, and so is zero. Hence
{y(a): y € G} consists of all the eigenvalues of T and the only
eigenfunctions are constant multiples of characters, using (3) of
§1. //
Theorem 3.4: (Representation Theorem)
An ergodic measure-preserving transformation T with pure point
spectrum is conjugate to an ergodic rotation on some compact abelian
group.
Proof: Let A = the group of all eigenvalues for T and give A
2
the discrete topology. If L (m) is separable then A is countable. In
the other case we shall need to use the character theory of groups
without a countable basis. Let G = Â, the character group of A.
G is compact and abelian. The map a : A -* K given by a(X) = X is
a homomorphism of the discrete group A and hence, by (2) of §5 of
Chapter 0, 3 a € G so that a(X)=X(a) X € A (where we write X
when we wish to consider "X" as a homomorphism of G to K).

69
Define S : G •*■ G by S(g) = ag. We claim that S is ergodic.
Suppose fS = f, f € L2(G), f = Hb.X., X. € A. Then the above
gives us that
Eb.X.(a)X.(g) <\- Eb.X.(g)
so, b.X.(a) = b.. But X.(a) = X. and therefore b.X. = b.. If
b. t 0 then necessarily X. = 1. Thus X.(g) = 1 for all g € G,
and we get that f = a constant a.e. We know then that S is
ergodic, and by the previous theorem has pure point spectrum.
Again by the previous theorem the eigenvalues of S =
{Y(a): y € G} = {a(X): X € A} = {X : X € A} = A. So, S and T have
the same eigenvalues and both have pure point spectrum. Hence the
Discrete Spectrum Theorem tells us that they are conjugate. //
Theorem 3.5: (Existence Theorem)
Every subgroup A of K is the group of eigenvalues of an
ergodic measure-preserving transformation with pure point spectrum.
Proof: The desired transformation is the rotation S
constructed in the proof of Theorem 3.4-. //
The conjugacy problem for ergodic measure-preserving
transformations with pure point spectrum is completely solved. We have some
very simple invariants, namely the eigenvalues, which determine when
two such transformations are conjugate. Also there are some simple
examples, namely group rotations, such that each ergodic measure-
preserving transformation with pure point spectrum is conjugate to
one of these examples. So each conjugacy class of ergodic measure-
preserving transformations with pure point spectrum is characterized
by a subgroup of K, and each subgroup of K corresponds to a
conjugacy class.

Chapter M- : Entropy
We are searching for conjugacy and/or isomorphism invariants.
In 1958 Kolmogorov [1] introduced the concept of entropy into ergodic
theory, and this has been the most successful invariant so far. For
example, in 194-3 it was known that the two-sided (1/2,1/2)-shift and
the two-sided (1/3,1/3)-shift both have countable Lebesgue spectrum
and hence are spectrally isomorphic; but it was not known whether they
were conjugate. This was resolved in 19 5 8 when Kolmogorov showed that
they had entropies log 2 and log 3 respectively and hence are not
conjugate. Von Neumann had had the same idea considerably earlier,
but he was unable to prove that entropy was a conjugacy invariant.
The notion of entropy now used is slightly different from that used
by Kolmogorov - the improvement was made by Sinai [1] in 1959.
§1. Partitions and Subalgebras
Throughout, (X,B,m) will denote a probability space.
Definition 4- .1 :
A partition of (X,B,m) is a disjoint collection of elements
of B whose union is X.
We shall be interested in finite partitions. They will be
denoted by Greek letters, e.g., Ç = {A,,...,A, }.
If Ç is a finite partition of (X,B,m) then the collection of
all elements of B which are unions of elements of Ç is a finite
sub-a-algebra of B. We denote it by A(ç). Conversely, if C is a
finite sub-a-algebra of B, say C = {C.: i=l,...,n}, then the
nonempty sets of the form B, fl . . . fl B where B. = C. or X \C. form
r J 1 n l i. l
a finite partition of (X,B,m). We denote it by Ç(C). Thus we have

71
a one-to-one correspondence between finite partitions and finite
sub-a-algebras of B.
Definition 4.2:
Suppose Ç and t| are two finite partitions. Ç ; tj means that
each element of Ç is a union of elements of r|.
Note: Ç ïti « A(ç) £. A(-q)
A £ C « £(A) s Ç(C).
Definition M-. 3 :
Let Ç = {A1,...,An}, ti = {C1,...,Ck}. Then
Ç v ti = {Ai n C. : 1 < i < n, lsjsk}.
If A and C are finite sub-a-algebras of B then A v C
denotes the smallest sub-a-algebra of B containing A and C.
Note: Ç(A v C) = Ç(A) v Ç(C)
A(Ç v ti) = A(Ç) v A(ti).
Suppose T: X -* X is a measure-preserving transformation. If
Ç = {A19...,Am}, then by T~nç we mean {T~nA1,...,T-nAm} and by
T~n(A) we mean {T~nA: A € A} (n > 0).
Note: If n > 0
ÇCT nA) = T nç(A)
A(T nO - T nA(ç)
T"n(A v C) = T~nA v T"nC
T"n(Ç v-q) = T nÇ v T-nT!
Ç £ ti ■=> T nÇ £ T_nT!
A £ C - T nA c- T nC.

72
Definition 4.4:
If V and E are (not necessarily finite) sub-a-algebras of B,
then we write OIE if V D € P 3 E € E such that m(DAE) = 0
and V E € E 3 D € P such that m(DAE) = 0.
If 0 and E are finite, 0 2 E, and if Ç(0) =
{Dl5...,D ,D +1,...,D } where m(D^) > 0 for 1 2 i £ p and
m(Di) = 0 for p+1 £ i £ q, then Ç(E) = {E^ . . . ,E ,E +1, . . . ,Eg}
where mtE^AD^) = 0 for 1 £ i £ p and m(E.) = 0 for p+1 £ i £ s.
V c C means V D € V 3 C € C such that m(DAC) = 0.
§2. Entropy
All logarithms are to base 2 and 0•log 0=0.
Let A S. B be finite. Let Ç(A) = {A., , . . . ,A, }. Then
k
H(A) = H(Ç(À)) = - E m(A.) log m(A.),
i = l
is called the entropy of A (or of Ç(A) ). (This means that if
A,,...,A, denote the outcomes of an experiment then H(A) measures
the uncertainty removed (or information gained) by performing the
experiment. H(A) is a measure of the uncertainty about which A.
a general point of X will belong to.)
Remarks :
(1) If A = {X,*} then H(A) = 0. Here A represents the outcomes
of a "certain" experiment so there is no uncertainty about the
outcome.
(2) If Ç(A) = {A1,...,A]<} where m(Ai) = 1/k V i then
k ,
H(A) = - E è log £ = log k.
i = l K K

73
Thus, we gain a lot of information if k is large. (Since all the
members of Ç(A) have equal measure there is much uncertainty about
which A. a point will belong to.)
(3) H(A) s 0.
(4) If A I C then H(A) = H(C).
Suppose T: X -* X is measure-preserving.
If A is a finite sub-a-algebra of B we define
h(T,A) = lim j^HCA v T_1A v ... v T~(n_1)A)
n-*»
n-1
= lim ±H( V T 1A),
n->- n i=o
which we call the entropy of _T_ with respect to _A_. (Later (in
Corollary 4.4) we will show that the above limit always exists.)
(This means that if we think of an application of T as a passage of
n-1 _i
one day of time, then \/ 1 k represents the combined experiment
i=0
of performing the original experiment represented by A on n
consecutive days. h(T,A) is then the average information per day that
one gets from performing the original experiment daily forever.)
Remarks :
(5) h(T,A) £ 0.
n-! _i n-! _i
(6) The elements of Ç( V T A) = V Ç(T A) are all the sets of
i=0 i=0
n-1 _.
the form (~\ T 1A where Ç(A) = {A,,...,A,}.
i=0 mi Ik
We define h(T) = sup h(T,A) where the supremum is taken over
all finite sub-a-algebras A contained in B and call this the
entropy of _J_. ( h(T) is the maximum average information per day
obtainable by performing a finite experiment.)

74
Remarks :
(7) h(T) 2: 0. h(T) could be +-.
(8) h(idv) =0. If h(T) = 0 then h(T,A) = 0 for every finite A,
A
n-l _±
which implies that V T A does not change much as n -* ».
i=0
Theorem 4.1:
Entropy is a conjugacy invariant and hence an isomorphism
invariant.
Proof: Let T, : X, -* X. , T„: X„ -* X„ be measure-preserving
and let $ : (8„,m„) -* (8..,m,) be an isomorphism of measure algebras
such that ST"1 = T"1*. Let A be finite, k^ c B , and £(A2) =
{A15.,.,A }. Choose B^ € B such that B^ = *(A.) and so that
r\ - {B.,...,B } forms a partition of (X,, B.. ,m.. ) . Let A. = A(t|).
n_1 -i
Now O T, B (where the q. € {l,...,r}) has the same measure
i=0 X qi
n-l _.
as Pi T„XA since
i=0 qi
♦(HA ) = *(OT:iA ) = Ht"1*(A ) = "fi T^B = ^ T^B .
i=0^1^qi i = 0 qi i = 0 qi i = 0 ~x ~qi i = 0^L^qi
n-l _. n-l _.
Thus, H( V T XA ) = H( V T„1A ) which implies that h(T ,A ) =
i=0 x x i=0 l L x x
h(T„,A„) which in turn implies h(T,) ï h(T„). By symmetry we then
get that h(Tx) = h(T2). //
Theorem 4.2:
The function 4> : [0,-) -* R defined by:
0 if x = 0
4>(x) =
I x.log x
if x ?! 0
is convex, i.e., 0(ax+py) £ a*(x) + p*(y) if x,y € [0,-), a,p > 0,
a+p = 1.

75
By induction
k k
*( Z a-;xJ - E a.-*Cx. )
i=l 1 1 i=l 1
k
if x. € CO,-), a. > 0, Z a. = 1.
1 1 i=l 1
Proof:
<t>' (x) = log e + log x
0"(x) = i£S_Ë. > o on (0,-).
x
Suppose y > x; by the mean value theorem
4>(y) - 4>(ax+{3y) = <t> ' (z)a(y-x)
where ax+py < z < y and
4>(ax+py) - 4>(x) = 4>'(w)p(y-x)
where x < w < ax+py.
Since 4," > 0 4>'(z) 2; *'(w), thus
pU(y) - 4>(ax+py)) = 0'(z)ap(y-x)
> 4> ' (w)ap(y-x) = a(*(x+py)-0(x)).
Therefore 4>(ax+{3y) 2 a^(x) + p*(y) if x,y > 0, and hence also if
x,y so by continuity of <t>. II
Corollary 4.2:
If Ç = {A1,...,A]<} then H(Ç) £ log k.
Proof: Put a. = 1/k and x. = m(A.) 1 £ i S k. Then
HCO s log k. //
Combined with remark (2) this corollary shows that among all the
partitions of X into k sets, the largest entropy is obtained when
all the sets have equal measure. This fits in with our intuitive
interpretation of entropy.

76
§3. Conditional Entropy
Let A,C £ B be finite.
5(A) = {A1,...,Ak}, ç(C) = {C15...,Cp}.
We define the entropy of A given C to be
p k m(A. PI C. ) m(A. fl C. )
H(A/C) = - £_ m(C.) Y ï 1- log ï 3_
j=l 3 i=l m(Cj) m(Cj)
m(A. fl C. )
= - 2. m(A. fl C . ) log J— 2: 0
i,j x 3 mCCj)
omitting the j-terms when m(C.) = 0.
So to get H(A/C) one considers C. as a measure space with
normalized measure m(-)/m(C.) and calculates the entropy of the
partition of C. induced by £(A) (this gives
k m^ fl C.) m(Ai fl C.)
- L ^— log 3— ) and then averages the answer taking
i=l m(Cj) m(Cj)
into account the size of C. ( H(A/C) measures the average
information obtained from performing the experiment associated with A
given the outcome of the experiment associated with C.)
Let W denote the a-field {*,X}. Then HCA/W) = H(A). (Since
N represents the outcome of the trivial experiment one gains nothing
from knowledge of it.)
Remarks :
(1) HCA/C) > 0.
(2) If A 2 V then H(A/C) = H(P/C).
(3) If C 2 V then H(A/C) = H(A/0).

77
Theorem 4.3 :
If A,C,P are finite subalgebras of B then:
(i) H(AvC/P) = H(A/P) + H(CMvp)
(ii) H(A vc) = H(A) + HCC/A)
(iii) A ^ C = H(A/0) s H(C/0)
(iv) A £• C = H(A) s H(C)
(v) C £. 0 = HCA/C) > H(A/0)
(vi) H(A) > H(A/0)
(vii) HCAvC/O) < H(A/0) + H(C/0)
(viii) H(A vC) s H(A) + H(C).
(ix) If T is measure-preserving then:
H(T"1A/T"1C) = H(A/C) and
(x) H(T_1A) = H(A).
(The reader should think of the intuitive meaning of each statement.
This enables one to remember these results easily.)
Proof: Let Ç(A) = {A.}, Ç(C) = {C.}, 5(0) = {D, } and assume,
x 3 k
without loss of generality, that all sets have strictly positive
measure (since if Ç(A) = {A,,...,A,} with rc(A.) > 0 1 < i 2 r
and m(A.) =0 r £ i £ k we can replace £(A) by
«Al Vl'N. UAr+l U ••• U V }"
m(A. n C. n D.)
(i) H(Av C/V) = - L_ m(A. fl C . n D, ) log ± J — .
i,5,k x 3 K m(Dk)
mCA.ncnD.) mCA.ncnD.) m(A. n D, )
But 3 £_ r i 3 !S_ i iS_ unless m(A. n D. ) = 0
m(Dk) m(AinDk) m(Dk)
and then the left hand side is zero and we need not consider it; and
therefore

78
m(A. fi D, )
H(AvC/0) = - E m(A. n C. n D ) log x K
i.j.k X 3 k m(Dk>
m(A. fl C fl D )
- I. m(A. fl C. fl D, ) log i 3 £-
i,j,k 3 mCA^ Dk)
m(A. fl D, )
= - L m(A. fl D, ) log i — + HCC/A v 0)
i,k 1 K m(Dk)
= H(A/P) + HCC/Av 0).
(ii) Put V - H = U,X}.
(iii) By (i)
H(C/0) = H(AvC/0) = H(A/0) + H(C/Avp) > H(A/0).
(iv) Put V = H in (iii).
m(D, fl C. ) m(A. fl D,)
(v) Fix i,j and let a = - ^—, x, = = —. Then
K m(C.) K m(Dk)
by Theorem 4.2
/ m(Dknc.) m(Ai n Dk) \ y m(Dk fl C. ) /itAJD^ \
\ k m(C.) m(Dk) J'y. m(C.) \ m(Dk) J
but since C £ V the left hand side
/m(A.nc.)\ m(A.nc) i(A.OC)
= 4, Ï 1- = i 3_ log i 2_ .
\ m(C.) J m(C.) m(C.)
Multiply both sides by m(C.) and sum over i and j to give
_ m(A. PC.) _ m(A. (1 D, ) m(A. fl D, )
2_ m(A. HO log 1- < 2_ m(D, n O — log 1
i,j J m(C.) i,j,k ^ J m(Dk) m(Dk)
_ m(A. n D, ) m(A. fl D, )
= 2. m(D. ) i — log i k_
i,k m(D,) m(D, )

79
or -HCA/C) S -H(A/Î>).
Therefore H(A/Î>) S HCA/C).
(vi) Put C = H in (v).
(vii) Use (i) and (v).
(viii) Set V - H in (vii).
(ix), (x) Clear from definitions. //
Theorem 4.4:
If {a } , satisfies a ï 0, a. sa +a V n,m, then
n n>l n n+m n m
lim a /n exists and equals inf a /n.
Ti~*-°° n
Proof: Fix m > 0. For each j>0 j=km + n where 0 5 n < m.
Then
a. a .. a a, a ka a a
j _ n+km < _n_ km ,. _n_ m _ _n_ , _m
j n + km km km km km km m
a • a a. a
As j -* » then k-*~ so lim — £ — and therefore lim -1 £ inf —.
j m j m
a a. a. a.
But inf — - lim —1 so that lim -J- exists and equals inf -1. //
m j 3 j
Corollary 4.4:
1 n"1 -i
If A £ B then lim - H( V T A) exists.
n-*- n i=0
(See also the remark after Theorem 4.6.)
n-1 _.
Proof: Let a = H( V T 1A) > 0.
n i=0

80
n+m-1
an+m = H< V T"lA)
1 = 0
n-1 _. n+m-1 _. by (viii)
< H( V T 1À) + H( V T 1À) „ „
. n „•_„ of Theorem 4.3.
i=0 l-n
m-1 _.
= a + H( V T A) by (x) in Theorem 4.3.
n i = 0
= a + a .
n m
We then apply Theorem 4.4. //
§4. Properties of h(T,A)
n-1 _.
Recall that h(T,A) = lim - H( V T 1A) .
n-*~ n i=0
Theorem 4.5:
Suppose A,C are subalgebras of B and T is measure-
preserving. Then
(1) h(T,A) < H(A).
(2) h(T,AvC) £ h(T,A) + h(T,C).
(3) A £ C = h(T,A) < h(T,C).
(4) h(T,A) < h(T,C) + H(A/C).
(5) If T is invertible and mil then
m
h(T,A) = h(T , V T1A).
i = -m
Proof:
(1) - H( V T 1A) s ± £ H(T 1A) by (viii) of Theorem 4.3.
n i = 0
n-1
i £ H(A) by (x) of Theorem 4.3.
= H(A).

81
(2) H( V T_i(A^O) = H( V T_iA v V T_:LC)
i=0 i=0 i=0
n-1 . n-1
£ H( V T_1A) + H( V T" C) by (viii) of Theorem 4.3.
i=0 i=0
(3) If A £. C then
n-1 . n-1
V T_1A £ V T_1C V n £ 1
i=0 i=0
so one uses (iv) of Theorem 4.3.
n-1 _. n-1 . n-1
(4) H( V T 1A) <H(( V T_1A) v ( y T_1C) ) by (iv) of Theorem 4.3
i=0 i=0 i=0
n-1 , n-1 . n-1 _,
= H( V T_1C) + H(( V T xA)/( V 1 C)) by (ii) of Theorem 4.3.
i=0 i=0 i=0
But, by (vii) of Theorem 4.3
n-1 . n-1 . n-1 . n-1 .
H((V T1A)/(V T_1C)) S Y. H(T_1A/(V T3C))
i=0 i=0 i=0 j=0
n-1
£ Z. H(T 1A/T 1C) by (v) of Theorem 4.3
i = 0
= nH(A/C) by (ix) of Theorem 4.3.
n-1 . n-1 _.
Thus, H( V T XA) < H( V T C) + nH(A/C).
i=0 i=0
m . , k-1 . m
(5) h(T , V T1A) = lim ^ H( V T"3 ( V T1A) )
-m k-*~ j=0 i = -m
, m+k-1
= lim i H( V T_:LA).
k-»~ -m

82
m+k-1 . m . m+k-1
H( V T_1À) = H( V T_1À v V T_1À)
-m -m m
m+k-1 • m . m+k-1
= H( V T_1A) + H(( V T_1À)/( V T 1A))
m -m m
by (ii) of Theorem 4.3
k-1 . m . m+k-1
H( V T_1A) + H(( V T 1A)/( V T 1A))
i=0 -m m
by (x) of Theorem 4.3.
We want to show that
, m . m+k-1
^H((VT:lA)/( V T 1A)) -> 0 as k->-.
-m m
, m . m+k-1 _• -, m .
But, i- H(( V T 1A)/( V T 1A)) < i- H( V T" A) by (vi) of Theo-
-m m -m
rem 4.3 and so the result follows. //
Theorem 4.6 :
If A £ B is finite, and T is measure-preserving then
h(T,A) = lim H(A/( V T 1A)),
n-*» i=l
Proof: The limit exists since the right hand side is non-
increasing in n by virtue of (v) of Theorem 4.3. We show by
induction that for n î 2
n-1 . n-1 j
H( V T 1A) = H(A) + Z H(A/( V T 1A)).
i=0 j=l i=l
This is true for n = 2 by (ii) of Theorem 4.3. Assume that this
equality is true for n. We shall prove that it holds for n+1.

83
H( V T_iA) = H( V T_1A) + H(A/( V T_1A))
i=0 i=l i=l
by (ii) of Theorem 4.3
n_1 -i n -i
= H( V T 1A) + H(A/( V T 1A))
i=0 i=l
by (x) of Theorem 4.3
= H(A) + Z H(A/( V T_iA)). (*)
j=l i=l
Dividing the above by n and taking the limit the result follows,
using the fact that the Cesaro limit of a convergent sequence is the
ordinary limit. //
Remark :
1 n_1 -i
± H( V T A) decreases to h(T,A).
n i = 0
n-1 . n
Proof: By (*) H( V T 1A) > nH(A/( V T 1A)) using (v) of
i=0 i=l
Theorem 4.3. Hence
n-1 . n . n-1
n[H( V T-1A) + H(A/( V T_1A))] s (n+l)H( V T_1A)
i=0 i=l i=0
n _. n-1
nH( V T 1A) < (n+l)H( V T 1A).
i=0 i=0
§5. Properties of h(T)
Theorem 4.7:
(1) For m > 0, h(Tm) = mh(T).
(2) If T is invertible then h(Tm) = |m|h(T) V m € Z.
Proof : We first show that
m-1
h(Tm, V T 1A) = mh(T,A).
i=0

84
This follows since
, k-1 . m-1 . km-1
lim ± H( V T ml( V T_:là)) = lim r^- H( V T 1A)
k-~ j = 0 i=0 k-~ Km i=0
= mh(T,A).
m-1
Thus, mh(T) = m • sup h(T,A) = sup h(Tm , V T_1A)
A finite A i=0
< sup hCI^.C) = h(Tm).
C
m-1 _.
Also, h(Tm,A) 2 h(Tm, V T 1A) = mh(T,A) by (3) of Theorem 4.5 and
i = 0
so, h(T ) £ mh(T). The result follows from these two inequalities.
(2) It suffices to show that h(T~ ) = h(T); and all we need to show
is that h(T_1,A) = h(T,A) for all finite A. But
n-1 . c _n 1 n-l
H( V T1A) = H(T"tn l; V TA) by (x) of Theorem i+. 3
i=0 i=0
= H( V T 3A). //
j = 0
Theorem 4.8:
Let (X,B,m) be a probability space and Bn be an algebra such
that the a-algebra generated by 8Q (denoted by CT(8Q) ) satisfies
aCB.) I B. Let C be a finite subalgebra of B. Then for every
e > 0, there exists a finite algebra 0, V £ BQ such that
H(0/C) + H(C/0) < e.
Proof: Let Ç(C) = {C,,...,C } and assume without loss of
generality that each C. has positive measure. We can do this since
if C,,...,C have positive measure and C .,...,C have zero
measure where 1 £ s < r then Ç(C') = {Cl9 . . .,Cg_1 ,Cg U ... U Cr>
is such that H(C'/0) = H(C/0) and H(0/C') = H(0/C).
Since *(x) = x-log x is continuous and 4> ( 0 ) = 0, <*>(!) = 0,

85
3 0 < 6Q < 1 such that -*(x) < e/2r if 0 £ x £ 6Q or
l-60 £ x £ 1.
We first show that if we choose a partition Ç(0) = {D-,,...,D }
m(C.)
3 m(C.AD.) £ min 6n — =6 V i, then H(C/P) < e/2. For, if
1 X lsi<r ° 2
Ç(0) satisfies the above inequality, then m(C. ) < m(D-) + 6 < m(D. ) +
m(C.)
which implies < m(D.) and hence,
2 !
m(Di) - m(Dinci) £ m(DiAC.) < 6 < 6Qm(Di).
m(D. fi C. )
Thus i — > l-6n.
m(Di) U
m(C. PI D- )
Therefore, if j i i then ^ — £ 6n and hence,
m(Di) U
v ,m(C. flD.)\
i j \ m(D.) /
S Y. m(.H.) T. e/2r = e/2.
i j
So m(C.AD.) < — min m(C) implies H(C/P) < e/2. If
1 2 l£i£r 1
60
m(C.iD.) £ -j— min m(C.) then this still holds and also m(C.iD-) <
1 l£isr
3
m(C.)/4 which implies m(D.) > r- m(C.) and hence
6 n - 6 n
m(C.AD.) < -7- ' T min m(D. ) < -~- min m(D. ).
11 4 d lsisr 1 ^ l£i£r 1
Therefore H(î>/C) < e/2 also.
So it suffices to show we can choose D. € 8. with
m(C.AD.) £ -£- min m(C.) = a.
4 l£i£r

86
Choose X > 0 3 X(r-l)[1+ r(r-l)] < a; for each i, choose
Bi € BQ 3 m(CiiBi) < X. If i i j then B± fl B. c ( B^C.^ U ( B. AC • )
so that m(B. (IB.) < 2X. Let N = IJ (B. FIB. ). We have
r-1
m(N) < r(r-l)\. Set D. = B.\N for 1 s i< r and D = X \ (J D..
i=l
{D,,...,D } is a partition of X and each D. € B.. If i < r then
D.AC, c (BiiCi) U N and so,
m(D.AC) < X[l+r(r-l)] < a.
r-1
However D AC c (J (D.AC.) and therefore
r i=l 1 1
m(DrACr) < (r-l)X[l +r(r-l)] < a.
So the theorem is proved. //
(If {A } is a sequence of subalgebras of B then V A de-
n ^ = n
n
notes the sub-algebra of B generated by the A .)
Corollary 4.8.
If {An} is an increasing sequence of finite algebras and
C c V An then H(C/A ) -> 0 as n -> -.
n n n
Proof: Let Bo = U Am 5 Bo is an algebra and C ° ct(Bq) by
m=l
hypothesis. By Theorem 4.8 V e > 0 3 finite V £ BQ 5 H(C/0 )< e.
But V -A for some m. since V is finite, so, if m î m„
H(C/A ) < H(C/A ) < H(C/0 ) < e.
m mn e
Thus, H(C/A ) tends to zero as m -* ». //
' m
For entropy to be useful we require methods of calculating its
value. The following is one of the main tools for calculating entropy.

87
Theorem 4.9: (Kolmogorov-Sinai Theorem)
Let T: (X,8,m) -* (X,B,m) be an invertible measure-preserving
transformation and let A be a finite subalgebra of B 3
oo
V TnA ? b. Then h(T) = h(T,A).
n = -»
Proof: Let C 5 B be finite. We want to show that
h(T,C) < h(T,A).
m m •
h(T,C) < h(T, V T1A) + H(C/ V TA)
i=-m i = -m
by (i+) of Theorem 4.5,
m
= h(T,A) + H(C/ V T1A)
i = -m
by (5) of Theorem 4.5.
m
Let Am = V TXA. It suffices to show that H(C/A ) goes to zero
i = -m m
as m -> ». This follows by Corollary 4.8. //
A similar result holds when T is not necessarily invertible:-
Theorem 4.10 :
If T: X -* X is a measure-preserving transformation (but not
necessarily invertible) and if A is a finite algebra contained in B
with V T1 2 B then h(T) = h(T,A).
i = 0
m-1 _.
Proof: This is similar to the previous theorem; use V T A
i=0
m .
in the place of V TA, and the formula
i = -m
m-1 _.
h(T,A) = h(T, V T 1A)
i = 0
(the proof of which is similar to (5) of Theorem 4.5). //
The following is sometimes useful in showing transformations

88
have zero entropy. We shall use it later to show a rotation of the
unit circle has zero entropy.
Corollary 4.10 :
If T is invertible and V T_1A % 8 for some finite A then
i=0
h(T) = 0.
Proof: By Theorem 4.10
h(T) = h(T,A)
n
= lim H(A/_V T_1A) by Theorem 4.6.
n-*» i=l
» . , n .
But V T-1A % T_1B = B. Let A = V T_:LA ; then k± c A c ...
i=l i=l
00
and V A % B. By Corollary 4.8 H(A/A ) - o and h(T) = 0. //
, n J J n
n=l
Remarks :
Entropy can be defined for any countable partition of (X,B,m)
as follows: If Ç = {A,,A,,...} then
H(Ç) = - I! m(Ai)log m(Ai)
(which may be infinite).
A countable partition Ç of X is called a generator for an
invertible measure-preserving transformation T if
ea
V TnA(Ç) 2 B.
n = --
As in Theorem 4.9, one can prove that if Ç is a generator and-
H(Ç) < - then h(T) = h(T,Ç).
The basic theorem on existence of generators Ç with H(ç) < »
was given by Rohlin in 1963. To state it we need the following
definitions: We say that (X,B,m) is countably generated if there exists
a countable collection {B } of elements of B such that the

89
cr-algebra generated by the {B } % 8. (X,B,m) is complete if every
subset of a set of measure zero is measurable. Suppose T: X -* X is
a measure-preserving transformation, and X is countably generated
and complete. We say that T is aperiodic if
m( \J {x € X: Tn(x) = x}) = 0.
n€Z
n*0
(The countably generated and completeness conditions ensure that this
set is measurable.) Note that T ergodic implies T is aperiodic
(unless the space is finite).
Theorem 4.11: (Rohlin [2], 1963)
Suppose (X,B,m) is a countably generated complete non-atomic
probability space and T: X -* X is an invertible measure-preserving
transformation. Then T has a generator Ç with H(Ç) < » iff
h(T) < » and T is aperiodic.
Thus, if T is ergodic and h(T) < » then T has a generator
Ç with H(Ç) < -.
Recently Krieger [1] proved:
Theorem 4.12:
If (X,B,m) is countably generated and T is an invertible
ergodic measure-preserving transformation such that h(T) < » then
T has a finite generator
Ç = {Al5...,An}.
In fact ç may be taken so that eh( < n < e + 1.
Hence finite generators exist in the most interesting cases
although they may be difficult to find.
We now prove some more results that are useful for computation
of entropy.

90
Theorem 4.13:
If BQ is an algebra and ct(Bq) g B then
h(T) = sup h(T,A)
where the supremum is taken over all finite subalgebras A of Bn.
Proof: Let e > 0. Let C £ B be finite. By Theorem 4.8 there
exists a finite V c Bn such that
e 0
H(C/0 ) < e.
e
Thus, h(T,C) < h(T,P ) + H(C/0 ) by (4) of
Theorem 4.5
< h(T,P ) + e.
e
Therefore, h(T,C) < sup h(T,0) + e,
V-BQ
V finite
and thus h(T) < sup h(T,0).
PSBQ
V finite
The opposite inequality is obvious. //
Theorem 4.14 :
Let A be finite subalgebras of B such that A, £ A £ ...
and V A ° B. Then h(T) = lim h(T,A ).
n n n
n=l n-*«
Proof : We note that h(T,A ) is an increasing sequence by (3)
ea
of Theorem 4.5. B = U A is an algebra and a(B ) ? B. By Theo
u n=l n u
rem 4.13 h(T) = sup h(T,C). If C ^ Bn is finite then C £ A
C£BQ ° n
C finit
for some n.. Thus
C£BQ u n0
h(T,C) s h(T,A )
n0

91
which implies h(T) < lim h(T,À )
n
n-v»
and hence h(T) = lim h(T,A ). //
n-*«
Theorem 4.15:
h(T1><T2) = h(Tx) + h(T2).
Proof: Let T : CX,^,!^) -* (X-^B^n^) and
T2: (X2,B2,m2) -> (X2,B2,m2).
If Ai - Bi' A2 £ B2 are finite then AixA9 """S fi-ni-te'
Ç(A1xA2) = {A1xA2: A1€Ç(A1), A2€Ç(A2)}.
Let Tn = U A,xA„
u A CB A co x *■
A1_e1,A2-e2
finite
= the algebra of finite unions
of measurable rectangles.
Thus, CTC'n^ " Bi*B2 ^y definition of 8,xB„, and by Theorem 4.13,
h(T,xT„) = sup h(TnxT„,C).
X A (iCT -1- *■
0
C finite
But if C is finite, C ç TQ then C £ A-xA. for some finite
A, ç; B,, A t B2. Hence
h(T,xT ) = sup h(T,xT,A,xA„).
1 l A!-Bi L I L I
A2iB2
A. ,A_ finite

92
n-1
H(y (T1xT2)~1(À1xA2))
i=0
n-1 _. n-1
= H(( V T,^) x ( y T^A.))
i=0 x i=0 L A
= - E (m1><m2)(CkxD- ) -log (m^nijXC^D.)
n-1 _.
where the C, are the members of Ç(V t7XA, ) ,
* i=0 1 1
n_1 -i
and the D. are the members of Ç ( V T„ A„)
3 i=0 l l
- - L m1(Ck)m2(D.)-log(m1(Ck)m2(D.))
= - E m1(Ck)m2(D.)-[log m^C^) + log m2(D.)]
= - E m1(Ck) -log n^CC^) - E m2(D-) -log m2(D-)
n-1 . n-1
= H( V T^AJ + H( V T^A.).
i=0 x x i=0 L A
Thus h(T1xT2,A1xA2) = h(T1,A1) + h(T2,A2)
so, h(T]_xT2) = h(Tx) + h(T2). //
§6. Examples
We shall now calculate the entropy of our examples.
(1) If I: (X,B,m) -*■ (X,B,m) is the identity, then h(I) = 0. This
is because h(I,A) = lim - H(A) = 0. Also, if Tp = I for some p^O
then h(T) = 0. This follows since 0 = h(Tp) = p-h(T) by
Theorem 4.7. Hence any measure-preserving transformation of a finite
space has zero entropy.
(2) Let T: K -> K be T(z) = az.
Case 1: Suppose {a : n € Z} is not dense, i.e. , a is a root of

93
unity. Thus a^ = 1 for some p t 0 ; and T^(z) = a^z = z so
h(T) = 0 by example (1).
Case 2: Suppose {an: n € Z} is dense in K. Then {a11: n<0} is
dense in K. Let Ç = {A.,A„} where
A. = upper half circle [1,-D
A„ = lower half circle [-1,1)
For n > 0 T~~ Ç consists of semi-circles beginning at a and
-a~n. Since {a : n > 0} is dense any semi-circle belongs to
V T_nAU). Hence any arc belongs to V T_nA(Ç). Thus,
n=0 n=0
ea
B = V T~nAU) and so, h(T) = 0 by Corollary 4.10. //
n=0
(3) Any rotation of a compact metric abelian group has entropy zero.
Proof : (a) Suppose X = Kn, the n-torus, and T(z,,...,z ) =
(a,z,,...,a z ). Then T = TnxT0x...xT where T. : K -* K is defined
linn il n î
by Ti(z) = aiz. By example (2) h(Ti) = 0 for all i so by
Theorem 4.15
n
h(T) = Y. h(T.) = 0.
i=l 1
(b) General Case. Let T: G -* G be T(x) = ax. Let G= {Yn sY, > • • • }•
Let H = Ker Yn H ... n Ker y • H is a closed subgroup of G and
n 1 n n
(G/H ) is the group generated by {y.,...,y }. Thus
(G/H ) = finite group * Z n,
n
i
so G/H = F x K n
n n
where F is a finite group and K is a finite-dimensional torus.
T induces a map T : G/H -*■ G/H by T (gH) = agH . T is a
c n n n-^n°n °n n

94
rotation on G/H , so that it can be written T = T , x T „ where
n' n n,l n, 2
T , is a rotation of F and T , is a rotation of K n. Thus,
n,1 n n, 2
h(T ) = h(T ,) + h(T „) = 0
n n ,1 n,i
by example (1) and case (a) of this proof.
Note that H \ {e} so that V A(G/H ) % B where A(G/H )
n n n
n
denotes the a-algebra consisting of those elements of B that are
unions of cosets of H .
Therefore, if B = \J A(G/H ) then by Theorem 4.13
u n=l n
h(T) = sup h(T,C).
C-BQ
C finite
However, if C £ BQ is finite then C £ A(G/H ) for some n and so,
h(T,C) < h(Tn) = 0. Thus, h(T) = 0. //
Corollary:
Any ergodic transformation with pure point spectrum has zero
entropy.
This follows from Theorem 3.4. (Actually we have shown the re-
2
suit only when L (m) is separable since the above calculation was
for a metric group G.)
(4) Endomorphisms of compact groups:
If A is an endomorphism of the n-torus K onto K we shall
show in Chapter 6 that h(A) = L. log|X| where the summation is over
all eigenvalues of the matrix [A] with absolute value greater than
one.
One can write down a complicated formula for the entropy of an
endomorphism of a general compact metric abelian group. See Juzvinskii
CI].

95
(5) Affine transformations:
We shall show in Theorem 6.10 that when T = a-A acts on Kn
then h(T) = h(A).
(6) The two-sided (p.,... ,p, _,)-shift has entropy
¥,.
P,- • log p..
i=0 1 1
Let X = IT {0,1,... ,k-l} and T be the shift. Let Ai =
— oo
i{x,}: xQ = i>, 0 < i < k-1. Then Ç = {AQ ,. . . >A]<-_1} is a partition
of X. Let À = À(Ç). By definition of B,
V TXA = B.
i=-»
By the Kolmogorov-Sinai Theorem (4.9),
h(T) = lim i H(À v T-1À v . . . v T~(n-1)A) .
n-*»
Atypical element of Ç ( A v T-1A v . . . v T-^-15 A) is
a. n t-1a. n ... n T~(n-1)A.
10 11 n-1
{{xn}: x0=iQ, ^1--i1,---^n.1--±n_1}
which has measure p. -p. ...-p. . Thus,
10 11 n-1
H(A vT-1A v ... vT"'""1^)
= - 11 (p. •••••Pn- )-log(p. -...-p. )
10 n-1 0 n-1
k-1
= - I! (P.: ■■•••Pi )[l0g p. + ... + log p. ]
in=...=i ,=0 x0 xn-l x0 n-1
0 n-1
k-1
= - n Y. Pi • log p • •
i=0 1

96
k-1
Therefore, h(T) = h(T,A) = - Z p.•log p.. //
i=0 1 1
Corollary:
The 2-sided (1/2,l/2)-shift has entropy log 2; the 2-sided
(1/3,1/3,l/3)-shift has entropy log 3. Thus these transformations
cannot be conjugate.
(7) The 1-sided (pQ,...,p, _.)-shift has entropy
k-1
- I Pi-log p-•
i=0 1 1
The proof is very similar to the one in example (6) but Theorem 4.10
is used instead of Theorem 4.9.
An example of a transformation with infinite entropy is the
following:
(8) Let I = (0,1] with Borel sets and Lebesgue measure. Let
oo
X = TT ! with product measure and let T be the shift on X. Then
h(T) = -.
To see this let A • = < {x. }: i^i < xn 5 - n>0 1 < i 5 n > .
n,i\l3Jn On J
Then m(A .) = 1/n and Ç = {A .,...,A } is a partition of X.
n,3. n n,x n,n
Hence h(T,£ ) = log n by the same argument as used in example (6)
-1 -k
(using the independence of Ç ,T Ç ,...,T Ç ). Therefore,
h(T) 2 log n for each n, and so h(T) = -. //
§7. How good an invariant is entropy?
Definition 4.5:
An invariant P for an equivalence relation is a complete
invariant if whenever T and S both have the property P then T
and S are equivalent.

97
Entropy is, in general, far from complet-».
(a) An example of two ergodic measure-preserving transformations with
equal entropy which are not conjugate.
Let T: K -* K be defined by T(z) = az, a € K, where {a11} is
dense in K, and let S: K -* K be defined by S(z) = bz, b € K,
where {bn} is dense in K. T and S are ergodic and h(T)= 0=h(S)
by example (2) of the previous section. If we choose a,b so that
{a11}"^ i {b }_^ then T and S are not conjugate (in fact, they
are not even spectrally equivalent) by Theorem 3.1. //
(b) An example (due to Anzai) of two ergodic and spectrally
equivalent measure-preserving transformations with equal entropy which
are not conjugate.
Let T: K2 -> K2 and S : K2 -> K2 be defined by
T(z,w) = (az,zpw) S(z,w) = (az,zqw)
n °° •
where {a }__ is dense in K and p,q are non-zero integers.
Observe that T and S are affine transformations, T and S are
ergodic, and h(T) = h(S) = 0 by example (5) of §6. By considering
2
the characters of K one can easily show that L (m) has a basis of
the form {gn: n 2: 0} U {U^f : j € Z, q i 0} where gn(T) = angn.
2 3
Similarly L (m) has a basis {g } U {Ugh : j € Z, q > 0}. One then
2 2
defines a unitary operator W: L (m) -* L (m) by W(g ) = g and
W(UTf ) = Ugh and extending. Clearly WUT = UgW showing T and S
are spectrally isomorphic.
However if p i +q T and S are not conjugate. As mentioned
before conjugacy and isomorphism coincide for measure-preserving
transformations of K equipped with completed Haar measure m. We
shall show T and S are not isomorphic. Suppose 4>T = S* and
4>(z,w) = (f ( z ,w) ,g( z,w) ) . f and g are only defined almost everywhere

98
but this will not affect our argument as we shall consider them as
members of L (m). We have f(T) = af and g(T) = fqg. Since f is
an eigenfunction with eigenvalue a, by remark (3) of §1 of Chapter 3,
f(z,w) = c*z for some c € K. The second equation then becomes
g(T(z,w)) =cqzqg(z,w). If one now expresses g as a Fourier series,
then it is straightforward to show that g(z,w) = kz w where k € K,
pm = q for some m € Z and a = c^. So 4>(z,w) = (cz,kz w ) is an
affine transformation and for <j> to be an invertible measure-preserving
transformation one needs m = ±1, i.e., p = ±q. //
However we can consider the problem of completeness of entropy
for certain collections of measure-preserving transformations, and
this we do in the next section.
§8. Bernoulli and Kolmogorov Automorphisms
(As general references for this section see Shields [2] and Friedman
and Ornstein [2].)
Definition 4.6:
Let (Y,F,|j.) be a probability space. Let
(X,B,m) = "(T (Y.F.m.)
and let T: X -* X be the shift
T({yn>) = {xn> where xn = yn+1 n € Z.
T is an invertible measure-preserving transformation and is called
the Bernoulli automorphism with state space (Y,F,m.).
Examples of Bernoulli automorphisms :
(1) the 2-sided (pQ,...,pk_1)-shift. Here Y = {0 ,1,... ,k-l} .
(2) the example (8) of §6. Here y = (0,1].
2
(3) If T is a Bernoulli automorphism so is T .

99
(i+) If T, and T„ are Bernoulli automorphisms so is T,xT
Remark :
If T is a Bernoulli automorphism then h(T) < » iff 3 a
countable partition -q on (Y,F,|j.) 3 H(t|) < » and A(t|) = F. In
this case h(T) = H(F).
We shall call a probability space a Lebesgue space if the
identity map of it is isomorphic (as a measure-preserving transformation)
to the identity map on a probability space consisting of a subinterval
of [0,1] with Lebesgue measurable sets and Lebesgue measure together
with some atoms.
Recently entropy has been shown to be a complete invariant for
the class of Bernoulli automorphisms:
Theorem 4.18: (Ornstein [1] and [2])
Two Bernoulli automorphisms (whose state spaces are Lebesgue
spaces) with the same entropy are conjugate; i.e., entropy is a
complete invariant for the conjugacy of Bernoulli automorphisms.
(Isomorphism follows from conjugacy by the assumption on the
state spaces.)
Certain special cases had been worked out earlier by Meshalkin
[1] and by Blum and Hanson [1]. This result reduces the conjugacy
problem for Bernoulli automorphisms to their state spaces, since the
entropy depends only on the state space. It is possible, for example,
for a Bernoulli automorphism with a state space of two points to be
conjugate to a Bernoulli automorphism with a countably infinite state
space.
Note :
Given any x > 0 one can find n > 0 and {p-,p„ ,...,p },
n n
p. £ 0 JT p. = 1 such that - £ p.-log p. = x. Hence V x > 0
1 i=0 1 i=0 1 1

100
3 a Bernoulli automorphism with entropy x.
Corollary 4.16:
(i) Every Bernoulli automorphism has an n-th root.
(ii) Every Bernoulli automorphism can be written as a direct product
of two Bernoulli automorphisms.
Proof: (i) Let T be a Bernoulli automorphism and n > 0. Let
S be a Bernoulli automorphism with h(S) = -h(T). Then Sn is a
Bernoulli automorphism with entropy h(T), and therefore S and T
are conjugate.
(ii) Let T be a Bernoulli automorphism. Let S be Bernoulli with
h(S) = Jj-h(T). Then h(SxS) = h(T) and since SxS is Bernoulli,
SxS is conjugate to T. //
Ornstein has proved many deep results about Bernoulli
automorphisms , some of which we now summarize :
Theorem 4.17: (Ornstein [3] etc.)
(i) Every root of a Bernoulli automorphism is a Bernoulli
automorphism. (S is an n-th root if Sn = T.)
(ii) Let T be a Bernoulli automorphism. If F is a sub-a-algebra
of B with TF = F then T restricted to the measure space
(X,F,m|r) is a Bernoulli automorphism (i.e., a factor of a Bernoulli
automorphism is a Bernoulli automorphism),
(iii) If F ? 8 is an increasing sequence of a-algebras with
TF = F and T restricted to (X,F ,m|E ) is a Bernoulli automor-
n n n r
n
phism V n > 0, then T is a Bernoulli automorphism (i.e., an
inverse limit of Bernoulli automorphisms is a Bernoulli automorphism).
The following class of transformations were introduced by
Kolmogorov in 1958 by analogy with regular stochastic processes.

101
Definition 4.7:
An invertible measure-preserving transformation T of a
probability space (X,B,m) is a Kolmogorov automorphism (K-automorphism)
if 3 a sub-a-algebra K of B such that:
(i) K c TK.
(ii) V TnK 2 B.
n=0
oo
(iii) O T"nK 2 W = {X,*}.
n=0
(If A,C are a-algebras A c C will mean V A € A 3 C € C with
m(AiC) =0. If B.jB, € B then B1 £ B2 means m(B1\ Bj) = 0 and
B. ? B„ means m(B,AB2) = 0.)
Theorem 4.18 :
Every Bernoulli automorphism is a Kolmogorov automorphism.
Proof: Let the state space for T be (Y,F,m.). If F € F, let
F
< {xn> €X: x € F > € B. Let G = {F: F€ F}, which is called the
0
time-0 a-algebra. Let K = V T1G. We now verify that K satis-
fies the conditions for a Kolmogorov automorphism.
0 . 1 .
(i) K = V T1G c V 1 G = TK.
i=-« i=-«
oo oo jl . oo ,
(ii) V TnK = V V T1G = V T1G = B by definition of B.
n=0 n=0 i=-» -»
00 00
(iii) We have to show f~) T~nK S H = {X,4>}. Fix A € P) T~nK =
0 0
oo -n oo
D V T1G. Let B € V T*G, j € Z. Since A € V T1G, A and
n=0 -" k=j i<j
B are independent, and therefore m(A(lB) = m(A)m(B). The collection
of all sets B for which m(An B) = m(A)m(B) is a monotone class,

102
and, by the above, contains U V T G. Therefore V B € B,
j=-- k=j
m(Af)B) = m(A)m(B). Put B = A, then m(A) = m(A)2 which implies
m(A) =0 or 1. Hence
H T"nK ? W. //
n=0
It was an open problem from 1958 to 1969 as to whether the
converse of Theorem 4.18 was true, i.e., whether a Kolmogorov
automorphism is conjugate to a Bernoulli automorphism. This was shown to be
false by Ornstein.
Theorem 4.19: (Ornstein [6])
There is an example of a Kolmogorov automorphism T with the
following properties:
(i) T is not conjugate to a Bernoulli automorphism,
(ii) T does not have a square root,
(iii) T is not conjugate to T~ .
Corollary 4.19:
Entropy is not a complete invariant for the class of Kolmogorov
automorphisms.
Proof: Let T be the example of Ornstein. 3y Theorem 4.9
h(T) > 0. Choose a Bernoulli automorphism S with h(S) = h(T).
S and T are not isomorphic by (i).
Remarks :
(1) Property (iii) of T contrasts with the behavior of ergodic
transformations with pure point spectrum. (See Corollary 3.3.)
(2) Ornstein's example is defined by induction and so is fairly
complicated to describe. It is therefore important to check whether the
more "natural" examples of Kolmogorov automorphisms are Bernoulli

103
automorphisms or not. We consider some of these at the end of this
section.
(3) Sinai has proved that if T is an ergodic invertible measure-
preserving transformation of (X,8,m), which we assume is isomorphic
to the unit interval with Lebesgue measure, and h(T) > 0 and if S
is a Bernoulli automorphism with h(S) £ h(T) then there exists a
measure-preserving transformation 0 such that 4>T = S*, i.e., S
is a factor of T.
The next theorem shows that Kolmogorov automorphisms and
Bernoulli automorphisms are spectrally the same:
Theorem 4.20: (Rohlin)
A Kolmogorov automorphism T of a probability space (X,B,m),
2
where L (m) is separable, has countable Lebesgue spectrum.
Proof: Assume B i {X,4>} = N or else the result is trivial.
We have (i) K ° TK (ii) V TnK ? B (iii) D T~nK ? W. We split
the proof into three parts :
a) We first show that K has no atoms, i.e., if C € K and m(C) >0
then 3 D € K with D c C and m(D) < m(C).
Suppose C is an atom of K with m(C) > 0. Then TC is an
atom of TK and since K c TK either TC £ C or m(CnTC) = 0. If
TC c C then TC % C since both sets have the same measure so that
C ç° OT~nK and therefore m(C) = 1. Hence B % {X,*}, a
contradiction. On the other hand, suppose m(TC (1C) = 0. Then either for some
k > 0 T C c C (and we use the above proof to get a contradiction)
or m(TkC n C) = 0 V k>0 and then C U TC U T2C U . . . has
infinite measure, a contradiction.
b) Let H = {f € L2(m): f is K-measurable}. Then U^H c H. Let
_ m .
H = V (6 UTH. From U~nH = © ujv (8 U^ ^ (n,m > 0) we conclude

104
that L (m) = (+) ILV © C where C is the subspace of constants. It
suffices to show V is infinite-dimensional since if {f.,f„ ,f.,...}
is a basis for V, then {fQ = 1, U-f : n €Z, m>0} is a basis for
L2(m).
o
c) We now show V is infinite-dimensional. Since TK t K (we are
o
assuming B t H) we know V t {0}. Let g € V, g t 0 and then
G = {x: g(x) i 0} satisfies m(G) > 0. Since G € K and using a)
we know XQH = {xQf: f i H} is infinite-dimensional. XQH = v' (ê XQUTH
where V c v so either V is infinite-dimensional (and hence V
is) or X„UTH is infinite-dimensional. In this case there is a
linearly independent sequence of functions {x„U_f } where the f
J bin n
are bounded functions in H. Then {gUTf } are linearly independent
in H. It suffices to show these functions are in V. But if f € H
then
(guTfn,uTf) = (g,uT(ffn)) = o
so gUTfn € V. //
Corollary 4.20:
A K-automorphism is strong mixing.
Proof: By Theorem 2.3. //
Kolmogorov automorphisms are connected to entropy by the
following result (half of which was proved by Pinsker).
Theorem 4.21: (Rohlin S Sinai [1])
Let (X,B,m) be isomorphic to [0,1] with Lebesgue measure.
Let T: X -* X be invertible and measure-preserving. Then T is a
K-automorphism iff h(T,A) > 0 V finite À % N; i.e., T has
completely positive entropy.

105
Remark:
This shows that K-automorphisms are "the opposites" of
transformations with zero entropy (since h(T,A) = 0 V A in the zero
entropy case).
Examples :
(1) Group Automorphisms. Rohlin proved that any ergodic automorphism
of a compact abelian metric group is a K-automorphism and later
Yusinskii proved the theorem in the non-abelian case. Katznelson [1]
has shown that ergodic automorphisms of finite-dimensional tori are
conjugate to Bernoulli automorphisms. Chu [l] and Lind [1] have
independently extended Katznelson's results to the (countably) infinite-
dimensional torus. Katznelson and Weiss [2] have also solved the
case where the dual group is the discrete rationals but whether an
ergodic automorphism of a general compact abelian metric group is
Bernoulli is not yet known.
(2) Markov Chains. Consider a two-sided Markov chain with transition
matrix [p..]. The shift T on the space of sequences of integers
becomes a measure-preserving transformation for the Markov measure
defined by tPi-;^ and an initial vector [p-i » • • • >Pk] satisfying
[pl5...,Pk][p^.] = Cpl5..-5pk]. (See Billingsley [1].) It is known
that T is ergodic iff the chain is irreducible (i.e. , V pairs of
states i,j 3 n > 0 with p.. > 0) and T is strong mixing iff
the chain is irreducible and aperiodic (i.e., V states i
g.c.d. {n: p.. > 0} = 1). Friedman and Ornstein [1] have shown
that if T is strong mixing then it is conjugate to a Bernoulli
automorphism. Therefore, from the point of view of ergodic theory mixing
Markov chains are the same as Bernoulli automorphisms, i.e., we can
represent the space as a direct product measure space so that T
becomes the shift on the new space.

106
(3) One can generalize the notion of a finite-dimensional torus to
obtain another kind of homogeneous space:--namely a nilmanifold. Let
N be a connected, simple connected, nilpotent Lie group and D a
discrete subgroup of N so that the quotient space N/D is compact.
N/D is called a nilmanifold. When N = Rn and D = Zn we get an
n-torus. The Haar measure on N determines a normalized Borel
measure on N/D. If A: N ■* N is a (continuous) automorphism with
AD = D then this induces a map A: N/D -* N/D, which we call an
automorphism of N/D. A always preserves the measure m. Parry has
investigated the ergodic theory of such maps and has shown that if A
is ergodic then A is a K-automorphism. A subclass of the
automorphisms of N/D, the Anosov ones, are known to be conjugate to
Bernoulli automorphisms, but it has not yet been proved that the others
are.
The simplest examples are as follows: Let
N satisfies the above conditions with the operation of matrix multi-
3
plication and the natural topology from R . Let
f/1 mpX
D = < 0 1 n : m,n,p € Z
l\0 0 1/
Then N/D is a nilmanifold. The automorphism
2
1 x+y
0 1
1 2x+y z+x +xy+^4-
of N induces an ergodic automorphism of N/D, and it is unknown if
this is Bernoulli.

107
§9. Pinsker Algebra
Let T be a measure-preserving transformation of a probability
space (X,B,m) which is isomorphic to [0,1] with Lebesgue measure.
Let
P(T) = V {A: A c B, À finite, h(T,A) = 0}.
This is called the Pinsker a-algebra.
One can show that T_1P(T) = P(T). One can also prove that if À
is finite then A c P(T) iff h(T,A) = 0. Thus, P(T) is the
maximum a-algebra such that T restricted to (X ,P(T),m|p,_.,) has zero
entropy. Note that P(T) = B iff h(T) = 0 and P(T) = N iff T
is a K-automorphism (by Theorem 4.21).
Theorem 4.22: (Rohlin)
If T is an invertible measure-preserving transformation with
h(T) > 0 then U™ has countable Lebesgue spectrum in
L2(B) 0 L2(P(T)).
This reduces the study of the spectrum of invertible measure-
preserving transformations to those with zero entropy.
Corollary 4.22:
Transformations with pure point spectrum have zero entropy.
2
Proof : L (m) cannot have a subspace on which U™ has countable
Lebesgue spectrum.
The types of spectrum that can occur for zero entropy
transformations are unknown. There are examples of zero entropy
transformations with countable Lebesgue spectrum (from Gaussian processes and
horocycle flows).
In the space of invertible measure-preserving transformations of
(X,B,m) with the weak topology, the set of transformations of zero

108
entropy forms a dense G,.
Pinsker [1] conjectured that any ergodic measure-preserving
transformation could be written as a direct product of one with zero
entropy and a K-automorphism. However, (ii) of Theorem 4.19 allows
us to obtain a counterexample for if T: X ■* X is the example of
Ornstein with no square root then the transformation S of the space
{0}xX|J{1>><x given by S(0,x) = (l,x). S(l,x) = (0,Tx) provides
a counterexample to the Pinsker conjecture. This example is not
2
mixing (since S is not ergodic), but Ornstein has constructed a
mixing transformation which violates the Pinsker conjecture.
§10. Sequence Entropy
(See: Kushnirenko [1].)
Let (X,B,m) be a probability space isomorphic to [0,1] with
Lebesgue measure. Let T: X -* X be an invertible measure-preserving
transformation.
Let r = {t.,t„,...} be a sequence of integers. Let A be a
finite algebra A <= 8.
Define hr(T,A) = lim sup ± H(T XA v . . . v T nA)
n-*~
and define hp(T) = sup hr(T,A).
A finite
It is easily shown that h„(T) is a conjugacy invariant for each r.
Entropy and spectral properties are connected by the following:
Theorem 4.23 : (Kushnirenko [1])
T has discrete spectrum iff h„(T) =0 V r.
One can also show that sup_h_(T) = » or log k, for some
k > 0, k € Z; moreover, those T with suprh„(T) = log k are easy
to describe.

109
Problem:
If T has quasi-discrete spectrum (see Hahn and Parry [1]), what
sequences give h„(T) > 0?
h_(T) has been calculated except in the cases when T has zero
entropy and r has "large gaps". h_(T) will only give new
information when h(T) = 0. (See: Newton [1].)
§11. Comments
Entropy was introduced as a conjugacy invariant for measure-
preserving transformations. It was soon realized that entropy was
more than just an assignment of a number to each transformation.
Kolmogorov automorphisms and transformations with zero entropy
have received the most treatment. They are "opposites" from the point
of view of entropy. Kolmogorov automorphisms are important for
applications as it seems that the most interesting smooth systems are
Kolmogorov and even Bernoulli.
By Theorem 4.22 we know that the spectral theory of invertible
measure-preserving transformations reduces to that for the zero
entropy case. The following is still an open problem:
If h(T) = 0 what kind of spectrum can UT have?
For transformations with zero entropy the isomorphism problem is
only solved for ergodic transformations with discrete spectrum,
totally ergodic transformations with quasi-discrete spectrum and some
other special cases. Sequence entropy may play a role in the
isomorphism problem for zero entropy transformations.
We note again that in the weak topology on the set of all
invertible measure-preserving transformations on a given space (X,B,m),
the set of transformations with zero entropy is a dense G.
(countable intersection of open sets) and the set of weak mixing
transformations is also a dense G.. Since the set of strong mixing

110
transformations is a set of first category it follows that "most"
transformations are weak mixing, have zero entropy, but are not strong
mixing.
The main problem to consider for Kolmogorov automorphisms seems
to be to find more examples of Kolmogorov automorphisms that are not
conjugate Bernoulli automorphisms. One should first check this fact
for all the usual ways of constructing new transformations from old
ones (e.g., is a weak mixing group extension of a Bernoulli
automorphism a Bernoulli automorphism?). Then one might hope to find a new
invariant that may be complete for Kolmogorov automorphisms.
§12. Non-invertible Transformations
Suppose T: (X,B,m) -* (X,B,m) is measure-preserving but not
necessarily invertible; assume that (X,B,m) is isomorphic to [0,1]
with Lebesgue measure. Note that
B :> T_1B :> T~2B ^ ...
Let B = Pi T_nB; so, T_1B = B , and T|,v „ , is invertible.
n=0 " " (X,B_,m;
One can show that U™ has countable Lebesgue spectrum in
2 2
L (B) 0 L (B_) where countable Lebesgue spectrum in this situation
means there is a basis of the form
{Unf : n>J and m >0}.
m
This reduces the study of spectral properties of measure-
preserving transformations to those which are invertible (in fact,
invertible ones with zero entropy by Theorem 4.22).
One can also show that P(T) c B (i.e., if h(T) = 0 then T
is invertible modulo sets of measure zero; more precisely T is a
measure algebra isomorphism).

Ill
The analogous concept to a K-automorphism is an exact endomor-
phism.
Definition 4.8:
T: X -* X is an exact endomorphism if
O T"nB 2 H ; i.e., B_ % N.
n=0
So exact endomorphisms are as far from being invertible as
possible. Examples of exact endomorphisms are the one-sided Bernoulli
shifts. Exact endomorphisms are strong mixing (by the above remarks
about spectrum and a proof like that of Theorem 2.3).
It was conjectured that every ergodic measure-preserving
transformation is a product of an exact endomorphism and an invertible
measure-preserving transformation. This is not so (Parry and Walters).
Also, one-sided Bernoulli shifts with the same entropy are not
necessarily conjugate since an m-to-1 map cannot be conjugate to an
n-to-1 map if m i n. So entropy is far from complete for exact
endomorphisms. Parry and Walters (1971) constructed two exact
endomorphisms S,T with S"nB = T_nB V n > 0, S2 = T2 (=. h(S) = h(T))
but with S and T not conjugate. The method used involved the
Jacobian of an endomorphism, a concept which was introduced in Parry
[3]. (It is not known if there are two K-automorphisms S,T with
2 2
S = T but with S and T not conjugate.) Also, exact
endomorphisms need not be conjugate to one-sided Bernoulli shifts; in fact a
one-sided Markov chain which is exact need not be conjugate to a
onesided Bernoulli shift.

Chapter 5 : Topological Dynamics
§0. Introduction
In measure theoretic ergodic theory one studies the asymptotic
properties of measure-preserving transformations. In topological
dynamics one studies the asymptotic properties of continuous maps.
Theorem 5.0:
Let X be a compact Hausdorff space. The following are
equivalent:
(1) X is metrizable.
(2) X has a countable base.
(3) C(X) (the space of all complex-valued continuous functions on X)
has a countable dense subset.
Proof: See Kelley [1].
We shall consider compact metric spaces X and homeomorphisms
T: X -* X. C(X) is a Banach algebra with
||f|| = sup |f(x)| .
x€X
The map UT: C(X) -> C(X), defined by (UTf)(x) = f(Tx) is clearly a
multiplicative linear isometry of C(X) onto C(X), i.e., IL, is a
Banach algebra automorphism.
Remarks :
Compactness is a "finiteness" condition and corresponds to the
assumption of a finite measure in the measure theoretic work. The
assumption of metrizability is not needed for many of the results but
it often shortens proofs and most applications are for metric spaces.

113
We assume that T is a homeomorphism, rather than a continuous map,
for simplicity.
Typical examples that we shall study are:
Examples :
(i) I on any X.
(ii) Tx = ax where X is a compact metric group. (On such a group
there exists a left invariant metric d, i.e.,
d(bx,by) = d(x,y) V b,x,y € X ).
(iii) an automorphism of a compact metric group.
(iv) an affine transformation Tx = a-A(x) where A is an
automorphism of a compact group G and a € G.
(v) Let Y = {0,1,...,k-l} with the discrete topology. Let
00
X = (f Y with the product topology. A metric on X is given by:
— ea
" Ix ~y I
d({x},{y}) = £ n , " . The shift T ( T{x} = {y } with
n n nr:. 2 ' '
y = x ., ) is a homeomorphism of X. Note that here we have a
Jn n+i c
special case of (iii) since X is a compact group under the operation
{xn} + {yn} = {(xn+yn)mod(k)},
and T is an automorphism of X. d is an invariant metric on X.
§1. Minimality
X will denote a compact metric space and. T : X ■*■ X a
homeomorphism. We would like to find a concept of "irreducible piece" to play
the role ergodicity played for measure-preserving transformations.
Definition 5.1:
T is minimal if V x € X {Tnx: n € Z} is dense in X.
0_(x) = {Tnx: n € Z} is called the T-orbit of x.

114
Theorem 5.1:
T is minimal iff TE = E and E closed =» E = * or X.
Proof: Suppose T is minimal and suppose E is closed, E i <t>
and TE = E. Choose x € E. Then 0T(x) c e by the T-invariance
of E, and X = 0_(x) so E = X i.e., E = X. Conversely, V
x € X, 0_(x) is a closed T-invariant non-empty set, and hence is
all of X. //
Definition 5.2:
A closed subset E of X which is T-invariant is called a
minimal set with respect to T: X -* X if T|_ is minimal.
Theorem 5.2 :
Any homeomorphism T: X •* X has a minimal set.
Proof: Let E denote the collection of all closed non-empty
T-invariant subsets of X. Clearly E i $ since X belongs to E.
E is a partially ordered set under inclusion. Every linearly ordered
subset of E has a least element (the intersection of the elements of
the chain. The least element is non-empty by Cantor's intersection
property.) Thus, by Zorn's Lemma, E has a minimum element. This
element is a minimal set for T. //
Remark :
Ergodicity has the properties:
(i) An ergodic transformation is indecomposable.
(ii) Every measure-preserving transformation on a decent space can be
decomposed into ergodic pieces in a nice way.
By its definition, a minimal transformation is indecomposable.
We know that each homeomorphism T: X •*■ X has a minimal set. Hswever,
in general, one cannot partition X into T-invariant closed sets E
such that X=Ue, TE =E V a and TI „ is minimal (although
a a a ' h °
a a

115
we can in some important cases). If T has such a decomposition it
is sometimes called semi-simple. An example of a transformation not
admitting such a decomposition is an ergodic automorphism of a compact
metric group. This will be clear from the next section.
Definition 5.3:
A point x € X is a periodic point of T if T x = x for some
n i 0. The least such positive n with this property is called the
period of x under T.
Theorem 5.3:
Let T: X •*■ X be a minimal homeomorphism. Then:
(1) T has no nonconstant invariant continuous functions.
(2) If X is not finite T has no periodic points.
Proof: (1) fT(x) = f(x) implies fTn(x) = f(x) V n € Z, and
so f is constant on the dense subset 0_(x) of X. Thus f is
constant on X.
(2) If Tnx = x, n i 0 then {x,Tx,...,Tn_1x} is a closed
T-invariant set and by the minimality condition it is the whole
space X. //
Remarks :
(i) If T has no nonconstant T-invariant functions then T need
not be minimal. As an example of this, let A be an ergodic
automorphism of a compact metric group G i {e}. Then A is not minimal
since A(e) = e. But A satisfies property (1), since if fA(x) =
f(x), f continuous, then by ergodicity, f = constant a.e. and,
since Haar measure is positive on open sets f is constant
everywhere .
(ii) The fact that a minimal homeomorphism of a non-finite space has
no periodic points rules out many important examples, such as ergodic
automorphisms of compact metric groups.

116
We now check whether the examples mentioned in §0 are minimal or
not.
Examples :
(i) I is minimal iff X = a single point.
(ii) Let G be a compact metric group and T(x) = ax. T is
minimal iff {a : n € Z} is dense in X.
Proof: ( = ) 0T(e) = {an : n€Z}.
(=) Let x € X. We must show that 0 (x) = X. Let y € X.
3 n- 3 a -* yx i.e. ,
n.
a -x -* y
ii
n.
T 1(x) - y.
Therefore 0„(x) is dense in X. //
(iii) An automorphism of a compact metric group G is minimal iff
G = {e}. The proof is trivial.
(iv) For affine transformations of compact metric groups necessary
and sufficient conditions for minimality are known. For example, if
G is also abelian and connected then T = a-A is minimal iff
O BnG = {e} and [a,BG] = G
n = 0
where B is the endomorphism of G defined by B(x) = x~ -A(x) and
[a,BG] denotes the smallest closed subgroup of G containing a and
BG. This was proved by Hoare and Parry [1],
(v) The shift on k symbols is minimal iff k = 0. This is seen
from (iii) above.

117
§2. Topological Transitivity
Definition 5.4:
T: X -* X is topologically transitive if 3 xQ € X 3 0T(xn)
is dense in X.
Note:
T minimal => T topologically transitive.
Theorem 5.4:
The following are equivalent:
(1) T is topologically transitive.
(2) TE = E, E closed, E i X => E is nowhere dense (or, equivalent-
ly, if TU = U, U open, U i <t> then U is dense).
(3) If U,V are nonempty open sets then 3 n € Z 3
Tn(U) fl V t 4,.
(4) {x € X: 0T(x) ?! X} is a set of first category.
Proof: (1) =» (2). Suppose 0T(xQ) = X and let E i «, E
closed, TE = E, E i X. Suppose U ç E is open, U i 4>. Then
3 p 5 Tp(xQ) € U £ E so that 0T(xQ) ç E and X = E, a
contradiction. Therefore E has no interior.
(2) =» (3). Suppose U,V i « are open sets. Then \J TnU is
an open T-invariant set; so, it is necessarily dense by condition (2).
Thus U TnU fl V i <t>.
n = -~
(3) => (4). Let Ul5U2,...,U ,... be a countable base for X.
Then 0^7x7 i X
» 3 n 5 0T(x) fl Un = <t>
~ 3 n 5 Tm(x) € X\U V m € Z
n

118
3 n with x € (~) Tm(X\U )
x € (J p| Tm(X\U ).
n=l m=-»
It suffices to show (~) T (X\U ) is a nowhere dense set V n. Its
m=-» n
00
complement is {J T (U ) which is clearly dense by condition (3).
m=— n
Hence the result follows.
(4) => (1). This is clear since a compact metric space is of
second category. //
The following theorem gives many examples of topologically
transitive homeomorphisms.
Theorem 5.5:
Let X be a compact metric space and T: X ■* X a homeomorphism,
m a Borel probability measure on X giving positive measure to every
non-empty open set. If T is ergodic with respect to m, then
m{x € X: 0T(x) = X} = 1. In particular, T is topologically
transitive .
Proof: Let U,,U„,... be a countable base for the topology.
By the previous proof
{x: 0T(x) i X} = U Pi TK(X\U ).
1 n = l k=— n
The closed set (~) T (X\U ) is T-invariant, so by ergodicity has
k=-- n
measure 0 or 1. But U is contained in complement of this set
and m(U ) > 0, since U is open. Therefore
n n c
ea
m( O Tk(X\U )) = 0
k=-
and so m{x: 0T(x) i X} = 0. Hence m{x: 0T(x) = X} = 1. //

119
Corollary 5.5:
Let G be a compact metric group and T: G -* G an affine
transformation. T is ergodic with respect to Haar measure «• T is topo-
logically transitive.
Proof: (=>) This is obvious as Haar measure is positive on
non-empty open sets.
(=) This proof is like the last part of the proof in example (5)
of §4 Chapter 1, which deals with the case when G is connected and
abelian. //
Theorem 5.6:
If T is topologically transitive then T has no nonconstant
continuous invariant functions.
Proof: Suppose f € C(X), fT(x) = f(x). If 0T(xQ) = X then f
is constant on 0_(xQ), a dense set, and hence is constant on X. //
Remarks :
(1) If all the T-invariant continuous functions are constant then T
need not be topologically transitive. The following is an example to
illustrate this:
Let X = K2x{0} U K2x{l} /
/ (e,0) ~ (e,l)
i.e., two copies of the two-torus joined at the identity. Let
2 2
A: K -* K be an ergodic automorphism and define T: X -* X by
T(x,0) = (Ax,0), T(x,l) = (Ax,l).
2
Then T is not topologically transitive since T preserves K x{0}
2
and K x{l}. However, each continuous T-invariant function is con-
2 2
stant since it must be constant on both K x{0} and K *{1},
because A is ergodic, and these two constants must be the same because
they must agree at the point (e,0) s (e,l). //

120
(2) T can be topologically transitive and have a dense set of
periodic points. To illustrate this we prove that any automorphism A
2
of K has a dense set of periodic points.
2
Fix n > 0. Consider the finite subgroup of K consisting of
points of the form (w,,w„) where w, = wn = 1. These are all the
2
elements of K of group order n. Since A is an automorphism it
preserves this finite subgroup and hence, each member of this subgroup
is a periodic point for A. If we now vary n we obtain a dense set.
This proof can obviously be extended to an automorphism of K ,
n > 1. //
(3) Topologically transitive homeomorphisms enjoy some of the
properties of minimal homeomorphisms and also allow other interesting things
to occur; e.g., a dense set of periodic points. (2) and (3) of
Theorem 5.4 show that topological transitivity is (in some sense) a
topological analogue of ergodicity. Also, topologically transitive
homeomorphisms are "indecomposable"; i.e., we cannot write
X = U E , TE = E and E closed
w a a a a
a
when T is topologically transitive. So it seems that topologically
transitive homeomorphisms are better building blocks than minimal
homeomorphisms. If T has a decomposition into minimal pieces then
each piece is also topologically transitive. So, the best thing to
do is to try to get a decomposition of T into topologically
transitive pieces, and then see if these pieces are also minimal.
A distal homeomorphism (i.e., x^y => 3 6>0 5
d(Tn(x) ,Tn(y)) > 6 V n € Z) can be decomposed into minimal pieces
(Ellis [1]). An Axiom A homeomorphism can be decomposed into
topologically transitive pieces (Smale [1]). But, not all homeomorphisms
can be decomposed into topologically transitive pieces; e.g., see the
example in remark (1) above.

121
The following gives a sufficient but not necessary condition for
a topologically transitive homeomorphism to be minimal.
Theorem 5.7 :
If X is a compact metrizable space, T: X ■* X a topologically
transitive homeomorphism, and if there exists a metric on X making
T an isometry, then T is minimal.
Proof: Suppose such a metric is d, i.e., d(Tx,Ty) = d(x,y).
Let 0T(xQ) = X and consider x € X. We want to show that 0_(x) =X.
Let y € X and let e > 0. There exist n,m € Z such that
d(x,Tm(xQ)) < e, d(y,Tn(xQ)) < e
so, d(y,Tn_m(x)) < d(y,Tn(xQ)) + d(Tn(xQ),Tn_m(x))
= d(y,Tn(xQ)) + d(Tm(xQ),x)
< 2e.
Therefore 0T(x) = X. //
We now check our examples for topological transitivity.
Examples :
(i) I is topologically transitive iff X = one point.
(ii) T(x) = ax is topologically transitive iff T is minimal iff
T is ergodic iff {a11: n € Z} is dense in X.
ni - ni
a -xQ ■* yxQ i.e., a
So, {a : n € Z} is dense in X. (Another proof would be to apply
Theorem 5.7 or Corollary 5.5.) //
(iii) An automorphism A of a compact metric group is topologically

122
transitive iff A is ergodic. (See Corollary 5.5.)
(iv) An affine transformation T of a compact metric group X is
topologically transitive iff T is ergodic. (See Corollary 5.5.)
(v) The shift on k symbols is topologically transitive. Consider
00
X = "U" {0,1,...,k-l}, T = shift. We know T is an automorphism of
— ea
the compact metric group X. The Haar measure on X is the measure
given by the weights 1/k,...,1/k. (To check this, fix x € X and
show, by checking on rectangles and using Theorem 1.1, that this
measure is invariant under translation by x.) T is ergodic with
respect to Haar measure, and therefore T is topologically transitive
by Corollary 5.5.
§3. Topological Conjugacy and Discrete Spectrum
When should we consider two homeomorphisms of compact spaces to
be the "same" from a dynamical point of view? The following seems
the most suitable:
Definition 5.5:
Let T: X -* X, S: Y -* Y be homeomorphisms of compact spaces.
T is topologically conjugate to S if there exists a homeomorphism
0 : X -* Y such that «T = S$.
Notes :
(1) This is an equivalence relation.
(2) If T and S are topologically conjugate then T is minimal
iff S is minimal and T is topologically transitive iff S is
topologically transitive.
Definition 5.6:
Let X be a compact metric space, T: X -* X a homeomorphism,
f a complex-valued continuous function on X. We say that f is an

123
eigenfunction for T if 3 X € C 3
f(Tx) = Xf(x) V x € X, and f i 0.
We then call X the corresponding eigenvalue for f.
Remarks :
Suppose T is topologically transitive.
(1) f(Tx) = Xf(x), f € C(X) - |\| = l and |f(x)| = constant.
Proof: f(Tx) = Xf(x) => |f(Tx)| = |x||f(x)|. Therefore,
sup |f(Tx)| = |X|sup |f(x)|
X€X x€X
and since T is onto = sup |f(x)|.
x€X
Therefore |X| = 1. Hence, |f(Tx)| = |f(x) | and by Theorem 5.6
|f(x)| = constant. //
(2) If fT = Xf, gT = Xg, f,g € C(X) then f = constant-g.
Proof: By (1), |g(x)| i 0 V x € X since g é 0. Therefore
(f/g)(Tx) = (f/g)(x) =» f/g = constant by Theorem 5.6. //
(3) Eigenfunctions corresponding to distinct eigenvalues are
linearly independent in C(X).
Proof: Let f (Tx) = X f (x) where {X } are all distinct for
n n n n
n = l,...,k. Suppose V x € X,
a1f1(x) + a2f2(x) + ... + 'akfk(x) = 0
where the a. € C for i = l,...,k.
By applying the above equation to T x instead of x, we get
alXlfl(x) + a2X2f2(x) + ••• + akXkfk(x) = ° V x € X-
Hence

124
/ 1
:v
,k-l ,k-l
Xl X2
a-f-(x)
a2f2(x)
£~7VkV*7 \°/
a, f, (x)
All the X. s are distinct so the matrix is nonsingular. Therefore
V
V x € X,
akfk(x)/ \o
i.e., a -f.(x)
a. = 0,
l
0 V x € X, i = 1,...,k, i.e
i
i = l,...,k since f- ^ 0. Hence, the f. s are linearly
independent in C(X). //
(4) The eigenvalues form a subgroup of the circle group K.
Under our assumptions T has only countably many eigenvalues.
To check there are only countably many eigenvalues it suffices to show
that if h: X -* K is an eigenfunction corresponding to an eigenvalue
t i 1 then ||h - 1|| > 1/4. For then two eigenfunctions, with values
in K, corresponding to different eigenvalues will be greater than
distance 1/4 apart and, since C(X) has a countable dense set,
there can only be countably many eigenvalues. So let h(Tx) = xh(x) ,
t i 1. Choose xQ € X and p so that xph(x ) is in the left-hand
half of the unit circle. Then
|h -1|| = sup ||h(x) -1||
x€X
ï ||h(Tpxn) -11
TPh(xQ) -1|| > 1/4. //
Definition 5.7:
Let T: X -* X be a homeomorphism of the compact metric space X.

125
We say that T has topological discrete spectrum if the smallest
closed linear subspace of C(X) containing the eigenfunctions of T
is C(X), i.e., the eigenfunctions span C(X).
Note:
When T is topologically transitive and has topological discrete
spectrum, 3 f : X -* X, n = 1,2,... linearly independent, spanning
C(X), such that f T(x) = X f (x). The following is a representation
n n n °
theorem for topologically transitive homeomorphisms with topological
discrete spectrum.
Theorem 5.8: (Halmos and von Neumann Cl])
The following are equivalent for a homeomorphism T of a compact
metric space X:
(1) T is topologically transitive and is an isometry for some metric
on X.
(2) T is topologically conjugate to a minimal rotation on a compact
abelian metric group.
(3) T is minimal and has topological discrete spectrum.
(4) T is topologically transitive and has topological discrete
spectrum.
Proof: (1) => (2). Let d be the isometry metric for T.
Suppose 0T(xQ) = X. Define a multiplication * in Om(xJ by
Tnx0*TmxQ = Tn+mxQ. We have
d(Tnxn*Tmxn,TPxn*Tqxn) = d(Tn+mxn,TP+qxn)
"O"-1- ^O'-1- "Q"1- "Q
< d(Tn+mx0,TP+mxQ) + d(TP+mx0,TP+qx0)
= d(Tnx0,TPxQ) + d(TmxQ,Tqx0),
Hence, the map * : 0T(xQ) xOT(xQ) -* 0_,(xQ) is uniformly continuous and
therefore can be extended uniquely to a continuous map a: XxX -» X.

126
Also, d(T nxQ,T mxQ) = d(Tmx0,TnxQ) and so,
„ , , inverse „ . .
0T(xQ) > 0T(xQ)
is uniformly continuous and can be uniquely extended to a continuous
map of X. Thus we get that X is a topological group and is also
abelian since it has a dense abelian subgroup {T xn: n € Z}. Since
T(TnxQ) = Tn+ xQ = Tx fcTnxQ we have Tx = TxQftx and so T is the
rotation by Tx0-
(2) ■» (3). if T is a minimal rotation on a compact abelian
group G then each character of G is an eigenfunction. Let A be
the collection of all finite linear combinations of characters. Then
A is a subalgebra of C(X) , contains the constants, is closed under
complex conjugation, and separates points. Applying the Stone-
Weierstrass Theorem we see that A = C(X).
(3) =» (4) is trivial.
(4) => (1). We can choose eigenfunctions f : X -* K, nil,
with f (T) = X f and where the f are linearly independent and
n n n n
span C(X). Define a metric on X by:
- |f (x) - f (y)|
d(x,y) = I —IL n
n=l 2n
- |\f (x) - \f (y)|
Then d(Tx,Ty) = I -^-2 —2-H = d(x,y).
n = l 2n
It remains to check that d gives the topology on X. If d(x ,x) -*0
then for all nil, as m -* »,
pr lfn(V - fn(x)l £ d(xm'x) "* °"
Thus, V nil, f„(x„) -* f (x) as m-*» and since {f„} separates
n m n n ' c

127
points, x -* x as m -* ». Conversely, suppose x -* x. Let e > 0,
and choose N such that
£ 4- < L-
n=N+l 2n 2
By the continuity of the functions f..,..., f„ 3 M 3 m > M =>
|fi(xm) - fi(x)| < e/2 i = 1,...,N. If m > M then
dUm'X) = i?i ? |fiUm) " fi(X)'
i=l 21 2 2
i.e. , d(xm,x) -♦ 0. //
Remark :
If Tx = ax is a minimal rotation of a compact metric abelian
group G it is straightforward to show that the set of eigenvalues
of T is {y(a): y € G} and every eigenfunction is a constant
multiple of a character. In fact, this follows from Theorem 3.3 since
2
each continuous eigenfunction is an L -eigenfunction.
We have the following isomorphism theorem.
Theorem 5.9: (Topological Discrete Spectrum Theorem)
Two minimal homeomorphisms of compact metric spaces both having
topological discrete spectrum are topologically conjugate iff they
have the same eigenvalues.
Proof: (1) The proof is along the lines of the proof of
Theorem 3.1, but instead of using Theorem 2.1 we use the Banach-Stone
Theorem. This says that if X,Y are compact spaces, $: C(Y) ■* C(X)
is a bijective linear isometry, and $(f-g) = *(f)*(g), then there
exists a homeomorphism ^ : X -* Y such that <Mf)(x) = f(^(x)).

128
(2) This theorem can also be proved using Theorem 5.8 and
character theory. By Theorem 5.8 we can suppose T is a minimal rotation
of a compact abelian group G, Tx = ax, and S is a minimal
rotation of a compact abelian group H, Sy = by. We are assuming
{Y(a): y € G} = {6(b): 6 € H}. Define a map 9 : H -* G by 9(6)(a) =
6(b). This is well-defined and a bijection. Moreover, 9 is easily
checked to be a group automorphism and hence induces an automorphism
C: G -* H. It is easy to show that CT = SC. //
Remark:
Thus the theory of topologically transitive homeomorphisms with
topological discrete spectrum is entirely analogous to that of ergodic
measure-preserving transformations with pure point spectrum.
§4. Invariant Measures for Homeomorphisms
In this section we consider some connections between the
topological and measure theoretic systems. We first prove some results
about Borel measures including the fact that a Borel measure on a
metric space is determined by how it integrates continuous functions.
By a Borel measure on X is meant a measure defined on the Borel
subsets of X, (i.e., the smallest a-algebra containing the closed sets).
Theorem 5.10:
A Borel probability measure m on a metric space X is regular
(i.e., if B denotes the Borel sets then V B € B and V e > 0 3
an open set U and a closed set C with C Ç B ç u and
m(U \C ) < e).
e e
Proof: (The proof does not require X to be metric but that
each closed set be a G..) Let R be the collection of all sets such
that the regularity condition holds, i.e., R={A€8:Ve>0 3
open U , closed C with C Ç A ç U and m(U\C)<e}. We show

129
that R is a a-algebra. Let A € R", we show that X\A € R. Let
e > 0. 3 open U , closed C with C Ç A £ U 3 m(U \C ) < e.
Thus, X\U ç X\A £ X\C and (X\C ) \ (X\U ) = U \C , so
e e eeee
m((X\Ce) \ (X\Ue)) = m(Ue\Ce) < e.
Therefore X\A € R.
We now show R is closed under countable unions. Let
00
A,, A ,... € R and let A = U A.. Let e > 0 be given. There
1 i=l 1
exist open U , closed C such that C £ A Ç u and
e,n e ,n e,n n e,n
ea
m(U \ C ) < e/3n. Let U = U U (which is open),
e,nxe,n e , e, n r>
n=l
~ k
C = U C , and choose k such that m(C \ I I C ) < e/2.
n=l n=l
Let C
e
= I J C (which is closed). We have C Ç A ç U . Also,
nVi e>n
m(UAC ) < m(U \C ) + m(C \C )
e e e e e e
nÇ1m(Ue,n\Ce,n) +m(£eXCe}
E
— + - = e.
n=l 3n 2
Therefore R is a a-algebra.
To complete the proof we show that R contains all the closed
subsets of X. Let C be a closed set and e > 0. Define
Un = {x € X: d(C,x) < 1/n}. This is an open set, U12 U2 2 ... 2 U 2 ..
ea
and O U. = C. Choose k such that m(U, \C) < e and let C = C
i=1 i k e
and Ue = Uk. This shows C € R. //

130
Corollary 5.10:
For a Borel probability measure m on a metric space X we have
that for a Berel set B
m(B) = sup m(C) and m(B) = inf m(U).
C closed U open
C Ç B U 2 B
Theorem 5.11:
Let m,|j. be two Borel probability measures on the metric
space X. If f dm = f d|i V f € C(X) then m = p..
Jx Jx
Proof: By the above corollary it suffices to show that m(C) =
M-(C) for all closed sets C £ X. Suppose C is closed and let
e > 0. By the regularity of m there exists an open set U, U 2 C
such that m(U\C) < e.
Define f: X -> R by
0 if x ^ U
d(x,X\U)
f(x) =
I. d(x,X\U) + d(x,C)
if x € U.
f is well-defined since the denominator is not zero. Also f is
continuous, f = 0 on X\U, f = 1 on C, and 0 < f(x) < 1 V
x € X. Then,
M.CC) < f f dp. = I f dm 5 m(U) < m(C) + e.
JX JX
Therefore m.(C) < m(C) + e V e > 0, so |j.(C) S m(C). By symmetry
we get that m(C) < |j.(C). //
Theorem 5.12: (Riesz Representation Theorem)
Let X be a compact metric space and J: C(X) -► C a continuous
linear map such that J is a positive operator (i.e., if f > 0 then
J(f) 5: 0) and J(l) = 1. Then there exists a Borel probability

131
= f f du
measure n on X such that
J(f)
for all f in C(X).
Proof: See Halmos [1], p. 247. //
The next theorem expresses the fact that the unit ball in the
dual space of C(X) is weakly compact.
Theorem 5.13 :
If {y. } is a sequence of Borel probability measures on a
compact metric space X, then there is a subsequence (Mv,} which
converges in the weak topology, i.e., 3 a Borel probability measure |j.
on X such that
[ f d^ - [ f d^
Jx ni JX
for all f in C(X).
Proof: We write |j.(f) = f du when f € C(X) and (i is a
JX
Borel measure. Choose f.,f2,... dense in C(X). Consider the
sequence of complex numbers {a ( f, ) } . This is bounded by l|f-,||, and
so has a convergent subsequence, say {|j. (f,)}. Consider
{p. (f„)}; this is bounded and so has a convergent subsequence
(2) (2 )
{p. (f„)}. Notice that {u: (f-,)} also converges. We proceed in
this manner, and for each i î 1, construct a subsequence {|j. } of
{|^n> such that i^^} £ {^1-1)} Ç ... Ç {|^(1)} £ {|^n>, and so that
{M. (f)} converges for f = f, ,f „ ,. . . ,f. . Consider the diagonal
{u:11 }. The sequence {|J. (f-)} converges for all i; thus
(m- (f)} converges for all f € C(X) (by an easy approximation

132
argument). Let J(f) = lim |j.nn)(f). Clearly J: C(X) -* C is linear
n-v»
and bounded, as |J(f)| £ ||f||. Also J(l) = 1, and if f > 0 then
J(f) a 0. By the Riesz Theorem, there exists a Borel probability
measure n on X such that J(f) = f du for all f € C(X), i.e.,
JX
f f d^n) - [ f d,x. //
Jy " Jy
X
Corollary 5.13:
The space of Borel probability measures on a compact metric
space X is itself a compact metric space under the weak topology.
Proof: Let f.,f„,... be dense in C(X). Define
- |f fi dm " f fi d^
D(m,n) = Z •
i = 1 2i||fil|
D is a metric en the space of Borel probability measures which gives
rise to the weak topology. The compactness follows from the previous
theorem. //
Theorem 5.14: (Krylov and Bogolioubov [1])
If T is a homeomorphism of a compact metric space X then
there exists a Borel probability measure on X which is preserved
by T.
Proof: Fix x € X. For f € C(X) and n £ 0, define
n-1
J„Cf) = =• T. f(T1(x)).
n n i = o
J : C(X) -* C satisfies the conditions of the Riesz Representation
n
Theorem (note that |j (f)| £ l|f||)3 so there exists a Borel
probability measure u. on X such that
J n

133
J (f) = I f du for all f € C(X),
n J n
By Theorem 5.13 there exists a subsequence {|j. } and a Borel
probability measure n on X such that
J (f) = [ f dp. + fd|i for all f € C(X).
nj Jx nj Jx
, n.
Since |j (foT) - J (f)| = — | fT D(x) - f(x)|
n3 n. n.
< ^-2||f|| -> 0 as j -> -
we have foT dp. = f d|x,
JX JX
i.e., J f d|^T-1 = f d|i V f € C(X).
jX JX
So by the uniqueness Theorem 5.11 for Borel measures we have that
M.(T-1B) = |j.(B) for all Borel sets B. //
Theorem 5.15:
Let T be a homeomorphism of a compact metric space X, and
let M„ denote the collection of all T-invariant Borel probability
measures on X (by Theorem 5.14, M_ i 0). Then
(1) MT is closed in the weak topology,
(2) MT is a convex set, and
(3) if m € Mm then m is an extreme point of Mm iff m is
ergodic with respect to T.
Proof: (1) Suppose {|j. } c M_ converges to |j. in the weak
topology. Then

134
J fT d^n "* J fT d
II
f f d,n - J f d,
so that |j. is T-invariant.
(2) is obvious.
(3) Suppose m € M™, m not ergodic. There exists a Borel
set E such that T~ E = E a.e. and 0 < m(E) < 1. Define measures
m, and m„ by
m ,„-, - m(B n E) an , m ,R, _ m(B n (X\E))
m. (B; = and m„(B; =
1 m(E) 2 m(X\E)
for B a Borel set. Note that m. and m„ are in M_, nu i m„,
and
m(B) = m(E)m1(B) + (1 - m(E))m2(B),
so that m is not an extreme point of M_.
Conversely, suppose m € M_ is ergodic, and
m = am. + (l-a)m„
where m.,m„ € MT, 0 < a < 1. We must show m. = m„. m. « m
(m. is absolutely continuous with respect to m) so that the Radon-
r dm.(x)
Nikodym derivative dm-,/dm exists, (i.e., m-.CE) = dm(x) ).
1 JE dm
-1 f aml
So, m, (E) = m.CT E) = —- dm(x)
1 ! J _x dm
T XE
f dmi -i .n
_L(T xy) dm(T Xy)
]■£ dm
f dml -1
—-CT y) dm(y).
J-p dm

135
dm, _, dm.
Therefore -j—— (T y) = g=—Cy) a.e.(m) (by uniqueness of the Radon-
Nikodyin derivative). But, m is ergodic, so that dm,/dm =
constant = k a.e.(m). Therefore
1 = m,(X) = J k dm = k-m(X) = k.
Jx
Since k = 1, m. = m and therefore m„ = m = m, . //
Definition 5.8:
T is uniquely ergodic if there is only one T-invariant Borel
probability measure on X, i.e., M„ = one point.
Remark:
T is uniquely ergodic with respect to m implies that T is
ergodic with respect to m.
Unique ergodicity is connected to minimality by:
Theorem 5.16 :
Suppose T is uniquely ergodic and m is its unique invariant
measure. T is minimal iff m(U) > 0 for all nonempty open sets U.
Proof: Suppose T is minimal. If U is open, U i ^ then
X = U Tn(U) ; so if m(U) = 0 then m(X) =0, a contradiction.
n=-»
Conversely, suppose m(U) > 0 for all open nonempty U. Suppose
also that T is not minimal, i.e., there exists a closed set K such
that TK = K, K i X. T|K has an invariant Borel probability
measure p.K on K by Theorem 5.14-. Define |j. on X by |j.(B) =
HK(K(1 B) for all Borel sets B. Then \i € MT and p. i m because
m(X\K) > 0 as X\K is nonempty and open while |j.(X\K) = 0. This
contradicts the unique ergodicity of T. //
The following results formulate unique ergodicity in terms of

136
ergodic averages.
Theorem 5.17:
The following are equivalent:
n-1 .
(1) V f € C(X), =• Y. f(T x) converges uniformly to a constant.
n i = 0
i n_1 i
(2) V f € C(X), =• Y. f(T x^ converges pointwise on X to a
n i = 0
constant.
(3) 3 m € MT 3 V f € C(X) and V x € X,
n-1
dm.
n . r
n iTo J
(4) T is uniquely ergodic.
Proof: (1) =» (2) holds trivially.
(2) = (3). Define k: C(X) ■* C by
1 n"1
k(f) = lim - Y. fT (x).
Observe that k is a linear operator and is continuous since
| J"!1 fT^x) | < |f|.
n i = 0
Also k(l) = 1 and f i 0 => k(f) ï 0. Thus by the Riesz
Representation Theorem there exists a Borel probability measure m such that
k(f) = I f dm. But k(fT) = k(f) and so, I fT dm = If dm, i.e.,
f dmT~ = f dm which implies that mT~ = m by 5.11, so that
m € MT-
(3) =» (4). Suppose that v € MT- We have
i F fTx(x) - f* V x,
n i=0
where f = f dm. Integrating with respect to v, and using the

137
bounded convergence theorem we get that
[ f dv = [ f* dv = f* = J f dm V f € C(X).
Hence v = m by 5.11. Therefore T is uniquely ergodic.
1 n_1 i
(M-) =• (1). If - ^ fT (x) converges uniformly to a constant
n i=0
then this constant must be f dm, where m is the unique T-invariant
measure. Suppose (1) is false. Then 3 g € C(X), 3 e > 0 3 V N
3 n > N and 3 x € X 3
n
n-1
dm I > e .
n" £0 gTi(V - J 8
n-1
1 *
Define J : C(X) -> C by J (f) = - F fT1(x ). J satisfies the
n J n n . _ « n n
conditions of the Riesz Representation Theorem. Hence, J (f) = f da
for some Borel probability measure |j. . Moreover, there exists a
subsequence {a } such that
i
Jn_(f) = J f d,n_ -. \ f d,
for all f € C(X) and for some Borel probability measure \i (by
Theorem 5.13). Then
| J (fT) - J (f)| = -jL |fT 1(xn ) - f(xn )|
ni ni ni ni ni
l
so that fT d(i = f dp..
Hence, p. € M_. But, | g dp. - g dm| > e so that p. i m
contradicting the uniqueness of m. //
We now see which of our examples are uniquely ergodic.

138
Examples :
(i) I is uniquely ergodic iff X = one point, since M_ = all
Borel probability measures.
(ii) T(x) = ax on a compact group is uniquely ergodic iff T is
minimal.
Proof: (=>) follows from Theorem 5.16 and the fact that Haar
measure is positive on open sets.
(«=) T is minimal «• {a11} is dense in G. Therefore G is
abelian. If 1 i y € G then
\ ¥ rC^x) = i "i rCaScx) - ^ (r(a)"-i) . 0 as n _
n i=o n i=0 n y(a) - 1
(note that y(a) i 1). So (2) of Theorem 5.17 holds when f is a
character and the condition (2) will hold for each f € C(X) by
approximation, since finite linear combinations of characters are
dense in C(X). //
(iii) An automorphism of a compact group G is uniquely ergodic iff
G = {e}, since Haar measure is preserved and so is the point measure
concentrated at e.
(iv) An affine transformation of a compact connected abelian metric
group is uniquely ergodic iff it is minimal.
Proof: (=>) follows by Theorem 5.16.
(=) follows, as in example (ii), by checking that (2) of
Theorem 5.17 holds. This was done by Hahn and Parry [1]. //
(v) The Bernoulli shift on k symbols is uniquely ergodic iff
k = 1. This is by example (iii).
An excellent survey of unique ergodicity and related topics can
be found in J. C. Oxtoby [1].

139
Recent results of Jewett [1] and Krieger [1] imply that any
ergodic invertible measure-preserving transformation of a Lebesgue
space is isomorphic in the sense of Chapter 2 to a uniquely ergodic
system. This indicates a certain lack of measure-theoretic import
for the concept of unique ergodicity. Hahn and Katznelson [1] have
found uniquely ergodic transformations in shift spaces with
arbitrarily large measure-theoretic entropies.

Chapter 6 : Topological Entropy
Adler, Konheim, and McAndrew [1] introduced topological entropy
as an invariant of topological conjugacy and also as an analogue of
measure theoretic entropy.
§1. Definition by Open Covers
All logarithms are to base 2. Let X be a compact topological
space. We shall be interested in open covers of X which we denote
by a,p, . . . .
Definition 6.1:
If a,p are open covers of X their join a v p is given by:
avp={AnB:A€a,B€p}.
Definition 6.2:
An open cover p is a refinement of an open cover a, written
a < p, if every member of (3 is a subset of a member of a. In
particular, a < a v p, {3<av(3.
Definition 6.3:
If a is an open cover of X and T: X -* X is continuous then
T~ a = {T~ (A): A € a} is an open cover of X.
Note:
T-1(a v p) = T-1(a) v T-1(p) and a < p => T-1a < T-1p .
Definition 6.4:
If a is an open cover of X let N(a) = the number of sets in
a finite subcover of a with smallest cardinality. We define the
entropy of a by: H(a) = log N(a).

141
Remarks :
(1) H(a) Ï 0.
(2) H(a) = 0 iff N(a) = 1 iff X € a. H(a) is small means that
there are a few sets in a which cover X. H(a) is large means that
some part of X is covered by a large number of sets in a and not
by a small number.
(3) a < p = H(a) £ H(p).
Proof: Let {B,,...>BN,g)} be a subcover of p with minimal
cardinality. V i 3 A. € a 3 A. 2 B.. So, {A, ,. . . ,Ay,, „, }
covers X and is a subcover of a. Thus N(a) £ N({3). //
(4) H(a v p) s H(a) + H(p).
Proof: Let {A, , . . . ,A.,, .. } be a subcover of a of minimal car-
1' ' N(a)
dinality, and {B, , . . . >BNf8^ } be a subcover of (3 of minimal
cardinality. Then
{Aj 0 B.: 1 £ i £ N(a), 1 < j < N(p)}
is a subcover of a v {3 so, N(a v |3 ) £ N(a)N((3). //
(5) If T: X -<• X is a continuous map then H(T~ a) £ H(a). If T
is also surjective then H(T a) = H(a).
Proof: If {A-, , . . . ,AN(. -, } is a subcover of a of minimal
cardinality then {T~ A,,...,T~ A.,, . } is a subcover of T~ a, so
N(T-1a) £ N(a). If T is onto, and {T-^, . . . ,T-1AN(T-ia) } is a
subcover of T~ a of minimal cardinality then {A, ,. . . ,A.,,T-i ■.}
also covers X, so N(a) £ N(T-1a). //
Theorem 6.1:
If a is an open cover of X and T: X -<• X is continuous, then
lim - H(a v T-1a v ... v T-(n-1)a) exists,
n— n

142
Proof: Recall that if we set
an = H(a v T-1a v ... v T (n 1}a)
then by Theorem M-.M- it suffices to show that:
a ï 0, and a. sa +a V nun.
n n+m n m
By (1), an ï 0, and
a +m = H(o v T_1a v ... v T-(n+m-:L)a)
n+m
£ H(a v T-1a v ... v T~(n-1)a)
+ H(T"na v ... v T-(n+m-:L)a) by U)
= a + H(T_na v ... v T"(n+m"1)
n
a)
a„ + H(T"n(a v ... v T (m"1)a))
n
an + H(a v ... v T (m-1)a) by (5)
= a + a . //
n m
Definition 6.5:
If a is an open cover of X and T: X ■* X is a continuous
map then the entropy of _J_ relative to _a_ is given by:-
h(T,a) = lim i. H(a v T-1a v ... v T~(n-1)a).
n-»-»
Remarks :
(6) h(T,a) i 0 by (1).
(7) a < p = h(T,a) < h(T,p).
n-1 . n-1 _.
Proof: a < p =» V T-:La < V T 1p , so by (3) we have that
i=0 i=0
H( V T a) £ H( V T p). Hence h(T,a) < h(T,p). //
i=0 i=0

143
Note that if {3 is a finite subcover of a then a < (3 so then
h(T,a) £ h(T,B).
(8) h(T,a) < H(a).
Proof: By (4) we have
H(a v T-1a v ... v T~(n-1)a)
n-1
< £ H(T a)
i = 0
< n-H(a) by (5). //
Definition 6.6:
If T: X ■* X is continuous, the topological entropy of T is
given by:
h(T) = sup h(T,a)
a
where a ranges over all open covers of X.
Remarks :
(9) h(T) 5: 0.
(10) In the definition of h(T) one can take the supremum over
finite covers of X. This follows from (7).
(11) h(I) = 0 where I is the identity map of X.
The next result shows that topological entropy is an invariant
of topological conjugacy.
Theorem 6.2:
If X,,X„ are compact spaces and T • : X. -<• X. are continuous
for i = 1,2, and are topologically conjugate, then they have the
same entropy.
Proof: Suppose ^: X, ■* X„ is a homeomorphism such that
*T1 = T2*" Let a ^e an °Pen cover °f X9' Then>

mk
h(T2,a) = lim i- H(a v T21a v ... v T2(n 1}a)
n
= lim i- H(«-1(a v T^a v ... v T~(n-1)a)) by (5)
n
= lim - HU-1a v T^1«"1a v ... v T^(n-1)0"1a)
n
= h(T1,«"1a).
By taking suprema the result follows. //
Adler, Konheim, and McAndrew proved several results about h(T).
In the next section we give a definition of topological entropy for
any uniformly continuous map of a metric space (not necessarily
compact). This definition can also be given for uniform spaces. The
definition will reduce to the previous definition in the compact case.
We shall prove the properties of h(T) with this new definition.
However, one result we would like to note is the following:
Theorem 6.3:
If T: X ■* X is a homeomorphism of a compact space X, then
h(T) = h(T-1).
Proof:
h(T,a) = lim - H(a v T-1a v ... v T~(n-1)a)
n
= lim i- H(Tn-1(a v T-1a v ... v T-(n-1)a))
n
n
= lim i H(a v Ta v ... v Tn-1a)
n n
= h(T-1,a). //
Adler, Konheim, and McAndrew could not answer the following
questions which they stated as conjectures in their paper.

145
Notation :
From now on the measure theoretic entropy of a transformation T
preserving a measure m will be written h (T).
r ° m
Conjecture 1:
Let X be compact and m a regular Borel measure on X. If
T: X ■* X is a homeomorphism preserving m then
hm(T) £ h(T).
m
[This has been proved by Goodwyn [1]. T need only be continuous. We
shall give a proof when X is a finite-dimensional torus. (See
Theorem 6.9.]
Conjecture 2 :
Let X be a compact metric space and T: X ■* X a homeomorphism.
(By Theorems 5.10 and 5.14 we know that MT, the set of Borel
measures on X invariant under T , is nonempty.) Then h(T) = sup h (T).
m€MT
[Partial contributions were made by Goodwyn [1] and Dinaburg [1], but
Goodman [1] finally proved it. We can drop the condition of X being
metric and allow T to be only continuous, provided we define M„ to
be all T-invariant regular Borel probability measures on X.]
Conjecture 3 :
If X is a compact metric group and T is an automorphism of X
then h(T) = h (T) where m is Haar measure,
m
[This was shown by Berg [1] and generalized by Bowen [i+]: T can be
an affine transformation and need not necessarily be invertible. We
shall prove this. (See Theorem 6.10.)]
Conjecture 4 :
Let {T.} be a one-parameter group of homeomorphisms of a
compact space X. Then h(T ) = | -t | hCTx).

146
[This was proved by Bowen when X is metric]
Conjecture 5 :
Let X,Y be compact spaces. Let {T : x € X} be a family of
homeomorphisms of Y so that
T(x,y) = (x,Tx(y))
is a continuous map of XxY. Then
h(T) = sup h(T ).
x€X x
[This was proved in the case where X and Y are metric by Bowen.]
§2. Bowen's Definition
If (X,d) is a metric space and x € X then B (x) will denote
the open ball centered at x and of radius e. UC(X,d) will denote
the collection of all uniformly continuous maps <t> : X -<• X.
Let T € UC(X,d); n € Z, n > 0; and e > 0. If K ç X, a
set F s X is said to (n,e)-span K with respect to T if V x € K
3 y € F such that
max d(T1x,T1y) < e.
0<i£n-l
For K compact, let r (e,K) be the smallest cardinality of any
(n,e)-spanning set for K with respect to T. We show later
(Theorem 6.4) that r (e,K) < -.
Set r_(e,K) = lim sup — log r (e,K).
n
A set E c x is (n,e)-separated with respect to T if, whenever
x,y € E, x ^'y then
max d(T1x,T1y) > e.
0si£n-l

147
For K compact, let s (e,K) denote the largest cardinality of
any (n,e)-separated subset of K with respect to T. We later show
(Theorem 6.4) that s (e,K) is finite.
Set sT(e,K) = lim sup — log s (e,K).
We define h(T,K) = lim rT(e,K) = lim sT(e,K). These limits
exist and are equal by Theorem 6.4. We then define
hd(T) = sup h(T,K).
K compact
Remarks :
(1) This definition can also be given in the context of uniform
spaces.
(2) h,(T) measures the amount of expansion in T (for the metric
d). For r (e,K) and s (e.K) to increase as n increases we need
n n
some expansion for T.
(3) The ideas for this definition come from the work of Kolmogorov
on the size of a metric space. If (X,p) is a metric space then a
subset F is said to e-span X if V x € X 3 y € F with
p(x,y) £ e, and a subset E is said to be s-separated if whenever
y,z € E, y i z, then p(y,z) > e. The e-entropy of (X,p) is then
the logarithm of the minimum number of elements of an e-spanning set
and the e-capacity is the logarithm of the maximum number of elements
in an e-separated set. So in the above definitions we are considering
the metric spaces (K,d ) where d is the restriction to the com-
n n
pact set K of the metric
p (x,y) = max d(T1x,T1y).
n 0£i<n-l
Then h(T,K) = lim lim sup — [e-entropy of (K,p )]. (It follows from
e^O n^- n n

148
the proof of the next theorem that to define h(T,K) it suffices to
consider spanning sets for K which are subsets of K.) sT(e,K) is
the average e-capacity of the spaces (K,d ) and h(T,K) is the
limit of sT(e,K) as e -* 0.
Theorem 6.4:
Suppose K is compact. Then
(i) r (e,K) £ s (e,K) £ r (e/2,K) < - and
(ii) if e.^ < e then r*T(e ,K) > rT(e„,K) and
IT(e1,K) ï sT(e2,K).
Proof: (i). We first show r (e,K) < ». There exists a 6 > 0
such that d(x,y) < 6 implies
max d(T1x,T1y) < e.
0£i£n-l
Then r (e,K) is less than the number of 6-balls needed to cover K
and hence is finite.
We shall now prove s (e,K) £ r (e/2,K). Suppose E ç K is an
n n
(n,e)-separated set and that F (n ,e/2)-spans K. Define 0 : E -* F
by choosing for each x € E some point 4>(x) € F with
max d(T1«(x),T1(x)) £ e/2. If *(x) = $(y) then
0£i£n-l
max d(T1x,T1y) £ e/2 + e/2 = e
0£i£n-l
so that x = y. Hence ^ is one-to-one and the cardinality of E is
less than or equal to the cardinality of F. Therefore s (e,K) £
rn(e/2,K).
Finally we show r (e,K)£s (e,K). Let E be an (n,e)-separated
subset of K of maximum cardinality. We claim that E (n,e)-spans
K, since if not 3 x € K 5

149
max d(T1x,T1y) > e V y € E.
0<i<n-l
Then E U {x} is an (n, e)-separated subset of K, contradicting the
choice of E.
(ii) is obvious. //
Hence the definition of h(T,K) makes sense.
Remarks :
(1) hd(T) depends on d.
(2) If K ç y. u .. . U Km are all compact then
h(T,K) £ max h(T,K.).
l£i£m
Proof: Certainly, s (e,K) £ s (e,^) + ... + sn<e>Km>' Fix
e > 0. V n choose K. (e) 5
n
s (e,^ (e)) = max s (e,K. ),
Then s (e,K) <m-s (e,K^ (e)) and so,
n
log s (e,K) £ log m + log s (e,K. (e)).
Choose n. -* » such that
— log s (e,K) ->■ lim sup — log s (e,K)
n. n. n n
and so that K. (e) does not depend on j (i.e., K. (e) = K(e)
n3 n.
V j). Thus, sT(e,K) £ sT(e,K(e)).
Choose e ->■ 0 so that K(e ) is constant ( = K. , say). Thus,
h(T,K) £ h(T,K. ) £ max h(T,K.). //
0 j 3

150
(3) V 6 > 0, in order to compute h, (T) it suffices to take the
supremum of h(T,K) over compact sets of diameter less than 6.
This is true by (2).
U) If X is compact, hd(T) = h(T,X).
Proof: By (2), if K c x, K compact, then
h(T,K) £ h(T,X). //
Definition 6.7 :
Two metrics d and d' on X are uniformly equivalent if
id. : (X,d) -> (X,d') and
id. : (X,d') -> (X,d)
are both uniformly continuous.
In this case, T € UC(X,d) iff T € UC(X,d').
Theorem 6.5:
If d and d are uniformly equivalent and T € UC(X,d) then
hd(T) = hdi(T).
Proof: Let e. > 0. Choose e„ > 0 5
d (x,y) < e„ => d(x,y) < e.
and choose e„ > 0 5
d(x,y) < e„ => d (x,y) < e„.
Let K be compact. Then
r (e1,K,d) £ rn(e2,K,d ) and
rn(e2,K,d') £ rn(e3>K,d).
Hence, r^Ce^K.d) £ rT(e2,K,d') £ ?T(e3,K,d).

151
If e, -* 0, then e„ -* 0, and e_ -* 0 so we have
hd(T,K) = hd.(T,K). //
Remark:
If X is compact and if d and d' are equivalent metrics then
they are uniformly equivalent. Also, each continuous map T: X -* X
is uniformly continuous.
Theorem 6.6: (Lebesgue Covering Lemma)
If (X,d) is a compact metric space and a is a finite open
cover of X then there exists a 6 > 0 such that each subset of X
of diameter s 6 lies in some member of a.
Proof: Let a = {A, ,...,A }. Assume the theorem is false. Then
for all n there exists B ç X such that diam(B ) < 1/n and B
n n n
is not contained in any A.. Choose x € B and select a subse-
^i n n
quence {x } which converges, say x •*■ x. Suppose x € A. € a.
Let a = d(x,X\A.) > 0. Choose n. such that n. > 2/a and
d(x ,x) < a/2. Then if y € B
n. J n •
l l
d(y,x) < d(y,x ) + d(x ,x) < — + - < a.
3 ni ni' ni 2
So y € A.. Hence B £ A., a contradiction. //
^ i ^
Theorem 6.7 :
When X is compact, Bowen's definition of entropy coincides with
the open cover definition.
Proof: For the duration of this proof let h*(T,a) and h*(T)
denote the numbers that occur in the open cover definition.
Let a - {A,,...,A } be an open cover of X. We shall show that
h*(T,a) S h(T). Let 6 be a Lebesgue number for a. Let F be a

152
(n,6/2)-spanning set for X of minimum cardinality. For z € F
choose A. (z),...,A. (z) in a so that B,,„(T z) £ A. (z). Let
x0 xn-l 6/2 xk
C(z) = A. (z) fi T-1A. (z) H ... H T~(n-1)A. (z),
0 1 n-1
which is a member of a v T~ a v ... v T ~ a.
We have X = U C(z) since if x € X 3 z € F 3
z€F
max d(T1x,T1z) £ 6/2
Osi<n-l
and hence x € T~k(Bg/2(Tkz)) ç l~kA± (z), 0 s k < n-1; so x €C(z).
k
Hence N(a v T-1a v ... v T~(n-1)a) < |F| = r (6/2,X), and
h*(T,a) S rT(6/2,X) £ h(T,X) = h(T)
since X is compact. Therefore h*(T) < h(T).
To prove the converse let 6 > 0 be given. Choose an open cover
a - {A,,...,A } of X such that diam(A.) < 6 for all i. Let E
be an (n,6)-separated subset of X with maximal cardinality. Two
members of E cannot belong to the same element of
a v T-1a v ... v T~ a since if
11-1 --i
x,y € Pi T JA. x,y € E
j = 0 xj
then max dCT^x^-^y) < 6 and so x = y.
0£j<n-l
So, N(a v T-1a v ... v T~(n-1)a) > |E| = s (6,X).
Therefore h*(T) > h*(T,a) ï s"T(6,X).
Letting 6 -<• 0 we have h*(T) > h(T,X) = h(T) . //

153
Notes :
(1) If we had set up the definitions using uniformities we would get
the above for compact Hausdorff spaces.
(2) Since a maximal separated set is spanning we get by the first
part of the proof of Theorem 6.7 that s (6/2,X) ï
N(a v T a v ... v t~ a) where 6 is a Lebesgue number for a.
Theorem 6.8:
(1) If T € UC(X,d) and m > 0 then h(Tm) = m-h(T).
(2) Let Ti € UC(Xi,di) i = 1,2. Define a metric on X1xX2 by
d((x1,x2) , (y1,y2) ) = max {d^x^y^ ,d2 (x2 ,y2) } . Then
hd(VT2) ïh^CV +hd2(T2).
If X, and X„ are compact then equality holds.
Proof: (1). Since r (e,K,Tm) £ r (e,K,T) we have
n mn
ilog rn(e,K,Tm) £ ^ log r^Ce ,K,T)
and therefore h,(Tm) < m-h,(T).
d d
Since T is uniformly continuous, V e>0 3 6>0 3
d(x,y) < 6 =» max dCT^x^^y) < e.
0£j£m-l
So an (n,8)-spanning set for K with respect to Tm is also an
(nm,e)-spanning set for K with respect to T. Hence, r (6,K,Tm) ï
r (e,K,T) so, m-r_(e,K) £ r _(6,K). Thus,
m-hd(T,K) < hd(Tm,K).
(2). Let K. ç x. be compact, i = 1,2. If F. is an (n,e)-
spanning set for K. with respect to T. then F,xF„ is an

154
(n,e)-spanning set for K,xK with respect to T,xT Hence,
rn(e,K1xK2,T1xT2) < rn(e,K1,T1)•rn(e,K2,T2)
which implies
rT xT (e,K1xK2) £ ?T (e,^) + ?T (e,K2).
Therefore
WWV - ^^l^^ +hd2(T2'K2}-
Let tt.: X,xx -* X. i = 1,2 be the projection map. If
K ç X1*x2 ^-s compact then K, = tt.(K) and K„ = tt„(K) are compact
and K ç ICxK Hence
hd(T1xT2,K) £ hd(T1xT2,K1xK2).
Therefore
h (T xT ) = sup h,(T xT ,K)
K£X1xX2 x L
compact
sup hd(T1xT2,K1xK2)
Kl-Xl
K2çX2
cpt.
sup h, (T,,K,) + sup h, (T„,K„)
Kl-Xl 1 K2çX2 a2 l l
cpt. cpt.
- h^CT^ + h^C^).
Now suppose X, and X„ are compact. Let a. be an open cover of
X. and have Lebesgue number 6. (i =1,2). If S. is a maximal
(n,6. /2)-separated set for X. with respect to T. then S-,xS2 is
an (n,6)-separated set for X-i*X2 with respect to T-i*T9 where

155
6 = min(61/2,62/2). Therefore
sn(6,X1xX2) 5: sn(61/2,X1)-sn(62/2,X2)
> N(a1 vT11a1 v . . . vT^11 1)a1)-N(a2 v T"1^ v . .. vT2(n 1)a2)
by note 2 above. Hence
h(T,xT„) > lim sup i log s (6,X,xX„)
1 I nn 1 I
lim - log N(a1 v T^a, v . . . v T (n "^O +
lim i- log N(a2 v T"1^ v . . . v T2(n 1}a2)
h(T1,a1) + h(T2,a2).
Since a1>a2 were arbitrary we get hCT.xT) > h(T.) + h(T2). //
Remarks :
(1) If T is a homeomorphism and T € UC(X,d), T-1 € UC(X,d) then
h (T) i hd(T-1) in general. We shall show later that if T: R -* R
is defined by T(x) = 2x then h(T) = log 2 while h(T~ ) = 0 using
the usual metric on R. (Note that T has expansion but T~ does
not.) However, on compact spaces h(T) = h(T~ ) (Theorem 6.3). This
is because on a compact space T has "as much expansion" as does T.
(2) Equality probably holds in (2) for non-compact X, and X„ but
I do not know a proof.
§3. Connections with Measure Theoretic Entropy
In this section we shall prove conjecture 3 (assuming
conjecture 1 is true) and we shall prove conjecture 1 when X is a

156
finite-dimensional torus.
Theorem 6.9: (Goodwyn)
Let X be a compact space and T: X -*■ X continuous. If m is
a T-invariant regular Borel probability measure on X, then
hm(T) £ h(T).
We shall prove this theorem when X is a finite-dimensional
torus since the proof is easier in this case.
k k k
Proof: Let X = K, T: K -<• K be any continuous map, and
let m be,any T-invariant Borel probability measure on K . We wish
to show that h (T) £ h(T).
m
Consider Kk as Rk/Zk with metric
d(x+Zk,y+Zk) = inf ||x-y+v|| x,y € Rk
v€Zk
where || • || denotes the usual Euclidean norm.
Fix an integer q > 0. Consider a decomposition of the unit
v
k-cube in R into all sets of the form
r PX PX+1 P2 P2+l
< (x, , . . . ,x, ) : — £ x, < , — £ x„ < ,
\ 1 k 2q 2q 2q 2q
pk pk+1
— £ x, < where
9q k oQ
0 < pi < 2q for i = 1,.. . ,k | .
This induces a partition of the torus K which we denote by
k 1
Ç = {A, , . , . ,A , }. Any ball in K of radius —t-=- intersects at
q l 2K1 2q
k k
most 2 members of Ç . Let y = {C,,...,C } be a cover of K by
open balls of radius —-pr. Let 2 6 be a Lebesgue number for y
For all x € Kk, fix some C(x) € Y with Bg(x) Ç c(x). Let F be

157
k .
an (n,6)-spanning set for K with respect to T of minimal cardi-
n-1 .
nality. Let x € Pi T A. , A. € ç . Choose y € F with
max dCT-'x.T-'y) £ 6,
0£j£n-l
and hence, TJx € C(TJy). Thus TJx € A. fl C(TJy). Hence if
15
Un = «V""'^-!^ npQ T"JAi3 * *>
then |Un| £ 2kn|F| = 2knrn(6,Kk). So, using Corollary 4.2,
Hm(Ç vT-1£ v... vT"(n-1)Ç ) £ log |Un| £ n-log 2k + log rn(6,Kk)
Thus, hm(T,Ç ) £ log 2k + lim i- log r (6,Kk)
q n n
= log 2k + ?T(6,Kk)
£ log 2k + h(T) = k + h(T).
(Note that we are taking logarithms to base 2.)
But, A(Ç ) / B as q ■* -. So, by Theorem 4.14
h (T) = lim hm(T,£ ) s k + h(T).
m m q
But this holds for any continuous T, so, in particular, for Tn,
n > 0. If n > 0
h (T) = - h (Tn) £ - [k + h(Tn)] = - + h(T)
m n m n n
so, by letting n ■* » we get the desired result. //

158
Theorem 6.10: (Bowen)
Let X be a compact metric group and T: X -* X, T = a-A an
affine transformation. If m denotes Haar measure on X then
h (T) = h(T) = h (A) = h(A).
m m
Proof: By the previous theorem h (T) £ h(T), and so it remains
r m
to prove that h(T) £ h (T). Suppose d is a left invariant metric
on X. Let B (y) = {x: d(x,y) < e} and
n-1 _, ,
D(x,e,T) = O T KB (T x).
n k=0 e
By induction we shall show that
T-kBe(Tkx) = x-(A"kBe(e)).
It is true for k = 0 by the invariance of the metric d. Assuming
it holds for k we prove it for k+1.
T-(k+l)B (Tk+lx) = T-1(T-kBe(Tk(Tx)))
= T-1(Tx-A-kBe(e))
- x.(A-(k+1)Be(e)).
n-1
Hence, D (x,e,T) = x • C] A B (e) = x-D (e,e,A).
n k=0 e n
Also, m(D (x,e,T)) = m(D (e,e,A)).
Let e > 0. Let £ = {A, ,...,A } be a partition of X into Borel
n-1 , n-1 _,
sets of diameter < e. If x ( f| T~ A. then O T A. S.
k=0 1k k=0 \
n-1 _, v v
x-D (e,e,A), since if y i C] T KA. then TK(x),TK(y) € A. V k,
n k=0 xk xk
and hence y € T~kB (Tkx) V k, i.e., y € D (x,e,T) = x-D (e,e,A).

159
n-1 .
Thus, m( O T~ A. ) < m(D (e,e,A)) and taking logs we see that
k=0 k n
n
F
(0 T"kA. ) log m( 0 T kA. )
i0'"-'in-l = 1 "k "k
n
YL m(PlTkAi ) log m(Dn(e,e,A))
io'--^in-r1
log m(Dn(e,e,A)).
Thus, hm(T) - hm(T'^) = lim n H(^ V '"" V T (n-1)£)
n
ï lim sup [- - log m(D (e,e,A))]
Hence, since e was arbitrary we obtain that
h (T) > lim lim sup [- - log m(D (e,e,A))] .
m e-*0 n n n
(The limit clearly exists.) Consider now an (n,e)-separated set with
respect to T, E ç X, having maximal cardinality. Then
U D (x,e/2,T) = (J x-D (e,e/2,A)
x€E n x€E n
is a disjoint union because of the choice of E. Therefore
s (e,X)-m(D (e,e/2,A)) < 1
and so s (e,X) £
m(Dn(e,e/2,A))
Therefore sT(e,X) S lim sup [- - log m(D (e ,e/2,A))]
n
and letting e •*■ 0 we see that

160
h(T) = hd(T,X) £ lim lim sup C- ± log m(D (e,e/2,A))]
e-*0 n
m
Thus, hm(T) = h(T) = lim lim SUP t_ n1 loS m(D (e,e,A))].
e-<-0 n
We can replace T by A here since the right hand side is
independent of a. //
Note:
The formula
h(T) = lim lim sup [- - log m(D (e,e,A))]
e-<-0 n
illustrates that T measures "the amount of expansion" in T.
§4-. Topological Entropy of Linear Maps and Toral Affines
Our aim in this section is to compute the topological entropy
(and hence by Theorem 6.10 the measure theoretic entropy) of affine
transformations of finite-dimensional tori. Recall (see §6 of
Chapter 0) that we can view the n-torus K either multiplicatively as
KxKx...xK (n factors) or additively as Rn/Zn. Each endomorphism A
of Kn onto Kn is given, in the additive notation, by
A(x + Zn) = [A]-x + Zn x € Rn,
where [A] is an n*n nonsingular matrix with integer entries. [A]
determines a linear transformation A of Rn and ttA = Att where
tt : Rn -* Kn is the natural projection given by tt(x) = x + Zn.
Let || • || denote the usual Euclidean norm on Rn. We define a
metric d on Rn/Zn by

161
d(x+Zn,y+Zn) = inf ||x-y+v|| x,y € Rn.
v€Zn
d is left and right invariant and, for every x € R tt maps the
ball of radius 1/4- about x in Rn isometrically onto the ball of
radius 1/4- about tt(x) in R /Z .
The next theorem deals with such a situation and asserts that
h,(A) = h~(Â) in this case, where d denotes the metric on Rn in-
a d
duced by the Euclidean norm || • ||. (Since ||Ax - Ay|| £ ||A|| • ||x - y|| we
know A € UC(Rn,d).) This will reduce the problem of calculating the
entropy of A to that of calculating the entropy of A.
Theorem 6.11:
Let (X,d),(X,d) be metric spaces and tt : X -* X a continuous
surjection such that there exists 6 > 0 with
"Ibr(2)! V*> -B6(n(x))
o
an isometric surjection for all x € X. If T € UC(X,d) and
T € UC(X,3) satisfy ttT = Ttt then
h,(T) = h~(T).
d d
Proof : If K is compact in X of diameter < 6 then tt(K) is
compact in X of diameter < 6. Every compact subset of X of
diameter < 6 is of this form. Let e > 0 be such that e < 6 and
if d(x,y) < e then d(Tx,Ty) < 6.
Suppose E ç. K is an (n,e)-separating set with respect to T.
We first prove that tt(E) is an (n ,s )-separating subset of tt(K)
with respect to T. To prove this, let x i y belong to E. Then
tt(x) i rr(y). Let iQ be chosen so that d(T x,T y) < e if i < iQ
- ~V1~ ~i0+1~
and d(T x,T y) > e. (This can be done since E is an (n,e)-
separating set with respect to T.) By our choice of e,

162
„ „i +1^ .i.+l^ ~i0+1~ ~i0+1~
d(T x,T y) < 6 and so T y € B (T x) which is mapped
V1 ~
isometrically onto BS(T tt(x)). So,
iQ+l „ iQ+l „ „ ~in+1~ ~in+1~
d(T U tt(x),T U rr(y)) = d(T U x,T U y) > e.
Thus tt(E) is (n,e)-separated with respect to T. Therefore,
s (e,K,ï) < sn(e,rr(K) ,T).
To prove the converse inequality, suppose E is an (n,e)-
separated subset of tt(K) £ X with respect to T, where K is
compact and of diameter < 6. Let È = tt~ (E) n K. Then È is an (n,e)-
separated set with respect to T since if d(T x,T y) £ e where
x,y € È then d( T1^ (x) ^tt (y) ) < e. Hence,
sn(e ,tt(K),T) £ sn(e,K,T).
Therefore s (e,K,T) = s (e,n(K),T)
and hence h„(T,K) = h,(T,rr(K)).
d d
By remark (3) of §2
h„(T) = h^(T). //
Corollary 6.11:
If A: Kn -<• Kn is an endomorphism then hH(A) = h~(A) where A
a d
is the linear map of Rn covering A, d is the metric on R
determined from the Euclidean norm and d is any metric on K .
We shall now proceed towards calculating the entropy of a linear
map of Rn.

163
Theorem 6.12:
Suppose A: Rp -* Rp is a linear map, and p a metric determined
by a norm on Rp. Then:
(i) h (A) > log |det A| if det A i 0, and
(ii) if all the eigenvalues of A have the same absolute value t
then
h (A) = max {0,p-log t}.
Proof: All norms on Rp are uniformly equivalent, so, by
Theorem 6.5 we can assume that p is the metric given by the Euclidean
norm. Obviously, A € UC(R ,p) as
p(Ax,Ay) = ||Ax -Ay|| £ ||A||||x-y|| = ||A||p(x,y).
(i). Let m denote Lebesgue measure on R . Then
m(A(E)) = |det A|-m(E)
for all Borel sets E Ç Rp. Let K £ Rp be compact and m(K) > 0.
If F (n,e)-spans K then K ç IJ D (x,e,A) = U [x +D (0,e,A)]
x€F x€F
n-1 . .
where D (x,e,A) = (~) A-1B (A x) (as in the proof of Theorem 6.10).
n i=0 e
Thus, m(K) < m(D (0,e,A))-r (e,K) = m(D (0,e,A))-r (e,K)
. „. m(K)
r (e,K) >
m(Dn(0,e,A))
Therefore r.(e,K) 5r lim ^- [log m(K) - log m(D (0,e,A))]
An n
n
= lim [- ^ log m(D (0,e,A))].
n = n
n
., _n m(B (0))
But, m(D(0,e,A)) < m(A ln l;B (0)) = —-T
n e i . . ,in-l
det A

164
so that rA(e,K) ï lim - [log | det A|n -1" - log m(B (0))]
n
= log |det A | .
Therefore h (A) ï h(A,K) > log |det A|.
(ii). In this case,
jdet Aj = |product of eigenvalues! = t".
So, by (i) h (A) ï p log t and then, clearly,
h (A) ï max {0 , p log t}.
We now have to show the opposite inequality. Assume first that
||A|| > 1. Let K be a compact subset of R^ of diameter < 1/2.
Choose b € RP such that 0 € b + K = K, . The diameter of K, < 1/2
b b
p
so that K, c I where
b 1
ij e {(x1,...,xp) € RP: |Xi| £ 1 V i}.
For 6 such that 0 < 6 < 1 let
F(6) = {(n16,...,n 6): n^ € Z, |ni6| < 2}.
Observe that |F(6)| £ (5/6)p, and 3 c > 0 5 V y € 1^, 3
x € F(6) 5 p(x,y) < c6. F(6) is an (n, ||A||nc6) -spanning set for
K, with respect to A, since if y € K, 3 x € F(6) 5
pCaSc.A^) £ ||Ai||p(x,y) < HAl^cS £ [|A||nc6,
for 0 < i £ n-1.
But then, F(6) - b is an (n,||A|| c6)-spanning set with respect to A
e
for K. Let e > 0, and set 6 = < 1, for n sufficiently
HA||n-c
large. Thus

165
rn(e,K) £
I e \ < I 5llAHnc V
l||A||nc/ ~ \ e )
for sufficiently large n. Also,
r.(e,K) = lim i log r (e,K)
a _ n n
n
£ lim 2- [log 5 + n log ||A|| + log c - log e]
n
P log ||A||
So, h (A) £ p log ||A||
and hence h (A) < max {0 , p log ||A||} if ||A|| > 1.
If ||A|| < 1 then h (A) = 0 since a (1 ,e )-spanning set is an (n,e)-
spanning set. Thus, in all cases
h (A) s max {0 , p log ||A[[}.
However, h (A) = - h (An) for n > 0,
P n p
< - max {0 , p log ||An|| }
= max {0 , p log [|An|[ n>.
But, ||An|| -+ the spectral radius of A, which here is precisely t.
Therefore
h (A) £ max {0 , p log t}. //
Remark :
If A: rP -<• rP is linear and the metric p is determined by a
norm then h (A) = lim lim sup [- — log m(D (0,e,A))] where m is
p e-<-0 n-*»

166
n-1 _.
Lebesgue measure on Rp and D (0,e,A) - (~) A-1B (0).
n i=0 e
Proof: We can suppose p is determined by the Euclidean norm.
In the proof of (i) of Theorem 6.12 we showed
r.(e,K) ï lim sup [- - log m(D (0,e,A))] and hence
a n n
n-*™
h (A) ï lim lim sup [- - log m(D (0,e,A))]. Let K be the p-cube
P e->0 n— n n q
in Rp with center 0 and side length 2q. If E is an (n,e)-
separated subset of K then {J D (x,e/2,A) is a disjoint union
q x€E n
and U D (x,e/2,A) = N x + D (0,e/2,A) ç K x. . Hence
xTe n xTe n q+2e
sn(e/2,K )-m(D (0,e/2,A)) £ (q+2e)P
and hence s.(e/2,K ) £ lim sup [- - m(D (0,e/2,A))].
A <1 n-*~ n n
Therefore h (T) = sup h (K ) < lim lim sup [- - m(D (0,e/2,A))]. //
P pq n nn
Theorem 6.13:
Suppose A: Rp -* Rp is linear and p is a metric coming from a
norm. Then
E ic
h (A) * *- log |\.|
where X,,...,\ are the eigenvalues of A.
Proof: By the Jordan Decomposition Theorem (Jordan Canonical
Form), we can write Rp as a direct sum of subspaces
RP = E, * ... ® E,
1 k
where A(E.) ç Ei for i = l,...,k and A. = A|£ has all its
i

167
eigenvalues with the same norm t.. Thus
and
A = A. (6 ... m A,
1 k
h (A) s f h (A.)
P ^i P !
by use of Theorem 6.8 and (since 6.8 is stated in terms of a specific
metric) the fact that all norms on R^ are equivalent. By
Theorem 6.12
k
h (A) £ Y. max f° > dim E- - loS T-^
= L. (dim E. • log t.)
T.>1 1 1
X
= E log |\.|.
|Xi|>l
We can suppose p is determined by the Euclidean norm. By the
above remark we have h (A) = lim lim sup [ log m(D (0 ,e ,A) )]. Write
e-<-0 n-*»
Rp as a direct sum of two subspaces Rp = F, ® F„ so that AF. Ç f.
(i = l,2) and A. = A|_ has eigenvalues with absolute value greater
1 tl
than one and A„ = A|_ has eigenvalues with absolute value less than
or equal to one. Since D (0,e,A) Ç B (0) PI A~ B (0) we have
ne e
m(D (0,e,A)) £ c Idet A~ I for some c independent of n.
n e ' 1 ' e r
Therefore lim sup [- - m(D (0,e,A)] ï log |det A11 = 2- log \\^\ .
Therefore
n-H- " " Uil>l
h (A) » I. log |\.j. //

168
Theorem 6.14:
Suppose T: K^ ■* K^ is an affine transformation, Tx = a-A(x)
where a € K^ and A is a surjective endomorphism of K^. If m
is Haar measure, then
h(T) = hm(T) = hm(A) = h(A) = L log |\i|,
the X.'s being the eigenvalues of the matrix [A] which
represents A.
Proof: We know by Theorem 6.10 that
h(T) = h (T) = h (A) = h(A)
m m
and by Corollary 6.11 that h(A) = h(A), where A denotes the
covering linear map of A. h(A) is calculated in Theorem 6.13. //
Note:
We have given a full proof of this result when the space is a
finite-dimensional torus (since we proved Theorem 6.9 only in this
case). The above proof is due to Bowen. This formula for the entropy
of an automorphism was first stated by Sinai [1].
§5. Expansive Homeomorphisms
As an analogue of the measure theoretic concept of a generator,
one could make the following definition:
Let (X,d) be a compact metric space, and T: X -* X a homeo-
morphism.
Definition 6.8:
A finite open cover a of X is a generator (weak generator)
for T if for every bisequence {A } of members of a,

169
O T Â is at most one point
' n
n=-~
(||TA is at most one point 1.
n=— n /
These concepts are due to Keynes and Robertson [1].
Theorem 6.15:
T has a generator iff T has a weak generator.
Proof: (=>) is trivial.
(«-). Let p be a weak generator for T,
p = {B1,...,Bs},
and let 6 be a Lebesgue number for p. Let a be a finite open
cover by sets A. having diam(A.) 5 6. So if A. is a bisequence
n
in a then V n 3 j 3 A. £ B. . Hence,
"""n •'n
n T~nÂi ç ô T_nB-
which is either empty or a single point. So a is a generator. //
The following shows that a generator determines the topology
on X.
Theorem 6.16:
Let a be a generator for T. Then V e>0 3 N>0 3 each
N
set in V T"na has diameter < s. Conversely, V N>0 3 e>0
-N
such that d(x,y) < e implies
N
x,y € OT"nA
-N n
for some A ..,...,A„ € a.
-N N

170
Proof: Suppose the first part of the theorem does not hold.
3 e>0 3 V j>0 3 x.,y., d(x.,y.) > e and 3 A. • € a,
3 3 J J J j1
A
„-i.
-j < i < j with x.,y. € (] T A. .. We can suppose that x. -* x,
3 3 i=-"i '1
y. •* y since X is compact, and hence x i y. Consider the sets
A. n. Infinitely many of them coincide since a is finite. Thus
x.,y. € A., say, for infinitely many j and hence x,y € A..
Similarly, for each n, infinitely many A. coincide and we
obtain A € a with x,y € T~n . Thus,
n ,J n
x,y € H T"nÂn
— ea
contradicting the fact that a is a generator.
To prove the converse let N > 0 be given. Let 6 > 0 be a
Lebesgue number for a. Choose e > 0 such that d(x,y) < e implies
d(T1x,T1y) < 6 for -N S i £ N. Hence if d(x,y) < e and |i| £ N
then T1x,T1y € A. for some A. € a. Hence
N
x,y € O T_1A.. //
-N
The .analogue of the Kolmogorov-Sinai Theorem is:
Theorem 6.17:
If a is a generator for T then
h(T) = h(T,a).
Proof: Let {3 be any open cover. Let 6 be a Lebesgue number
for p. Choose N > 0 so that each member of V T~ a has diameter
N "N
< 6. Then p < V T na, and so,
-N

171
N
h(T,p) < h(T , V Ta)
-N
, n-1 . N
= lim 4 H( V T_1( V T~na))
k— i=0 -N
, N+k-1
= lim 4 H( V T"na)
k-~ K -N
, 2N+k-l
= lim ~ H( V T~na)
k-- K 0
,„., . . 2N+k-l
- n. 2N+k-l 1 „/ \/ _-n v
■ iiz —*~' 2N+kTïH( Y
= h(T,a).
Therefore, h(T,{3) £ h(T,a) for all open covers p. Hence
h(T) = h(T,a). //
Remark:
The same result holds for weak generators.
Generators are connected with the notion of expansive homeomor-
phism, which was studied long ago.
Definition 6.10:
A homeomorphism T: X -* X is expansive if 3 5 > 0 5 if
x ?! y then 3 n € Z 5 d(Tnx,Tny) 5 6. We call 5 an expansive
constant for T.
Remark :
Another way to define an expansive homeomorphism is as follows.
Consider X*X with TxT acting on it. Define a metric D on XxX
by D((u,v),(x,y)) = max {d(u,x),d(v,y)}. Then T is expansive «»
3 6 > 0 such that if (x,y) is not an element of the diagonal,

172
some power of TxT takes (x,y) out of the 6-neighborhood of the
diagonal.
The following theorem is due to Reddy, and Keynes and Robertson.
Theorem 6.18:
T is expansive iff T has a generator iff T has a weak
generator.
Proof: By Theorem 6.15 it suffices to show T is expansive
iff T has a generator.
(-) Let 6 be an expansive constant for T and a a finite
00
cover by open balls of radius 6/2. Suppose x,y f f) T A where
A € a. Then, d(Tnx,Tny) £ 6 V n, so, by assumption x = y.
Therefore a is a generator.
(«■) Conversely, suppose a is a generator. Let 6 be a Le-
besgue number for a. If d(Tnx,Tny) < 6 V n then V n 3 A € a
3 Tnx,Tny € A and so,
ea
x,y € H T"nAn
which is at most one point. Hence x = y and T is expansive. //
Corollary 6.18 :
(1) Expansiveness is independent of the metric (however, the
expansive constant does change).
(2) T is expansive iff T is expansive, k i 0-
(3) Expansiveness is a topological conjugacy invariant.
Proof: (1) This is trivial, since having a generator has
nothing to do with the metric.
(2). If a is a generator for T then
a v Ta v ... v T~^k~ a

173
is a generator for T . If a is a generator for T then a is
also a generator for T.
(3) is trivial. //
The next result shows how to find measure theoretic generators
for expansive homeomorphisms.
Theorem 6.19 :
Let T be expansive with constant 6- If
Ç = {C1,...,Cs}
is a partition of X into Borel sets of diameter < 6, then Ç is a
measure theoretic generator for any T-invariant Borel probability
measure.
Proof: Let C. be a bisequence of members of Ç. If
n
x>y € H T~nc- then Tnx,Tny ç Ci for all n, and hence
-» n n
ea
d(Tnx,Tny) < 6 V n. By expansiveness x = y. Thus P)T~nC. = *
-» n
or = one point. Hence
ee
V TnA(ç) = B. //
— ea
Thus, for expansive homeomorphisms there are many measure
theoretic generators.
Examples :
(1) Isometries are never expansive except on finite spaces.
(2) Let A be an automorphism of the n-torus, and [A] the
corresponding matrix. Then A is expansive iff [A] has no eigenvalues
of modulus 1.
Sketch of proof: One first shows that A is expansive iff the
linear map A of R that covers A is expansive. Then show that

174
A is expansive iff the complexification of À is expansive. Then
one shows that the complexification of A is expansive iff the
transformation given by the Jordan normal form is expansive. Lastly, one
shows that the normal form is expansive iff there are no eigenvalues
of modulus 1.
(Note: By Theorem 6.19, any partition of Kn into sufficiently small
n-rectangles is a measure theoretic generator for an expansive
automorphism of Kn.)
(3) The two-sided shift on k symbols is expansive.
Proof (1): Let the state space be {0,1,...,k-l}. Let
Ai = |{xn>: xQ = i|, i = 0,1,...,k-l. Then AQ U A± U ... U Ak_± = X
and each A. is open. a - {A-.,. . . ,A, -, } is a generator for the
shift since if x € (~\ T~nAi where the A. € a then
-» n n
X = (•••,l_2'1_T51Q51T512'''
We then use Theorem 6.18. //
Proof (2): Let d be the metric given by:
|x - y
d({x >,{y }) = Z
n -1 n'
n Jn nr- 2'n'
Suppose {x } i {y }. Then for some nn, x i y and
n n u n. n.
n„ n
d(T °{x>,T °{yn}) - Z -err l*r
y„
n"~ "n" n = _- 2|n| n+n0 n+n0
lxn " yn
n0 n0
Thus 1 is an expansive constant. //
Remarks :
(a) If T: X -* X is expansive and Y is a closed subset of X with

175
TY = Y then T|„ is expansive.
(b) If T,: X, -* X,, T„: X„ -* X„ are expansive then so is
TjXT,: X,xX„ -* X,xX„. Any finite direct product of expansive homeo-
morphisms is expansive, but infinite products are not.
(c) If T: X -* X is expansive and S : Y -* Y is a homeomorphism
with
T
X >-X
* [ [ *
Y >Y
commutative for <t> : X -* Y a continuous surjection then S need not
be expansive. S is expansive if ^ is a k-to-one covering map. So
expansiveness is not preserved under the operation of taking factors.
Example : (Parry and Walters)
2 - -
Consider the 2-torus K ; identify (z,w) with (z,w). The
map <t> is a 2-to-one covering map except at four points:
(1,1), (-1,1), (-1,-1), and (1,-1).
2 2 . . .2
Let A: K -* K be an automorphism. The quotient space is S and
2
A induces a homeomorphism on S which can be shown to be non-
expansive .
Another example can be constructed as follows:
Let Tz = az be a minimal rotation of K. We shall represent T
as a factor of a subset of the two-sided shift on two symbols.
Consider the cover of K by the closed intervals (arcs) between -1 and
1 on K. Call one of them AQ and the other A,.
n n f ^*\ A
If z € K\{a ,-a : n € Z} we can uniquely associate n
a member of ~[T {0,1} to z by
z
iely associate / \ 1
Tnz € A . Let A denote the subset of Tf {0,1}
n -00

176
arising in this way. The map
*: A -> K \{an,-an: n € Z}
defined above satisfies tf>S(x) = Ttf>(x), x € A where S denotes the
shift, and if we can show <t> is uniformly continuous then <t> extends
uniquely to a continuous map tt : A -* K with ttS = Ttt . Suppose s > 0
is given. Choose N > 0 so that {l,a~ ,a~ ,. . . ,a~ } is e/2-dense
in K. Suppose {b } and {c } are two members of A such that
b = c for |n| s N. We then have to show d(tf> ( {b }) ,<t> ( {c }) ) < e.
Let x = tf>({b }) and y = +C{c }). The assumption b = c , |n| < N
means that anx and any belong to the same element of the cover for
|n| i N. If y = -x then this clearly cannot happen. So suppose the
counter-clockwise distance from y to x is smaller than the
clockwise distance. For some n with |n| £ N anx is in the open
interval of length e starting at 1 and going counter-clockwise.
Hence a y must also be in the upper half of the circle and by the
assumption about the relative positions of x and y, ay must be
between 1 and anx. Hence d(a x,a y) < e and so d(x,y) < s.
We shall now show that every expansive homeomorphism is a factor
of a subset of a two-sided shift.
Theorem 6.20:
Let T: X -* X be an expansive homeomorphism. Then 3 an
integer k > 0, a closed subset 2 of
oo
xk = TT{o,i,-.-,k-i}
_ oo
such that ct2 = 2, where a is the shift on X, , and a continuous
surjection tt : 2 -* X such that
rra(y) = Trr(y) y € 2.

177
Proof: The proof will resemble that of the preceding example.
Let 6 be an expansive constant for T. Choose a cover a =
{A.,...,A, _.} by closed sets such that diam(A.) < 6 V i and so
that the A. intersect only in their boundaries. Let D denote the
union of the boundaries of the A.. Then D„ = U TnD is a first
category set and so X\D_ is dense in X. For each x € X\D_ we can
00 • n
assign, uniquely, a member of X, by x -* {a }_„ iff T x € A .
n
Let A denote the collection of all sequences arising in this way.
If * : A -* X\D_ is the map defined above then *a(y) = T*(y) V y €A
and if we can show $ is uniformly continuous it will then follow
that tf> can be uniquely extended to a continuous map tt : A ■* X such
that rra(y) = Trr(y) V y « Î.
Let s > 0 be given. Choose N > 0 so that each member of
N
V Tna has diameter less than s, which can be done by Theorem 6.16,
-N
since we can enlarge each A. to an open set to obtain a generator
(remembering diam(A.) < 6). If {a >,{b } € A and a = b for
l n n n n
N
|n| £ N then *({a }), ({b }) are in the same member of V Tna
n -N
and so d(tf>{a },*{b }) < e. Hence <t> is uniformly continuous. //
The periodic points of T are associated with a generator as
follows :
Theorem 6.21:
Let T: X -* X be expansive and let a be a generator (or a weak
generator). Then T x = x iff
x = O Tk,1(A0 n ta1 n ... n t Ak_1)
i = -»
where A. € a j = 0,...,k-l.

178
k k-1
Proof: Suppose T x = x. Since a vTo v ... vl a is a cover
k-1 t
of X, x € An n TA. n ... n T A, . for some collection of A. s
' 0 1 k-1 -i
:
in a. Thus
x € Tk,1(A0 H TA1 H ... H Tk_1Ak_1) V i,
and this implies
x € P) Tk'X(AQ n TA1 fi ... H Tk_1Ak_1)
which is at most one point. Therefore,
x = n Tk-:L(A0 n ta n ... n Tk-1Ak_1).
The converse is trivial. //
The following gives an estimate on the number of fixed points
of Tn.
Corollary 6.21:
If T is expansive and a is a generator for T with M
members, then
Nn(T) = |{x: Tnx=x}| < Mn (n > 0).
Topological entropy is connected to periodic points by
Theorem 6.22 :
If T is expansive, then
h(T) > ca(T) = ïïm - log N (T).
n— n n
n-1 _.
Proof: Let a be a generator for T. Each element of V T~ a
0
contains at most one point fixed by T , since if Tx=x, Ty=y

179
n-1
and x,y € P) T-1A. then
i=0 :i
x,y € O T"nk(A. n t_1a. n ... n t"^"1^. )
k=-- 30 :1 ^n-l
which is at most one point. Therefore x = y. Thus,
N (T) s N(a vT_1a v . . . v T-^-1^)
n
which implies that
i log N (T) ; - H(a vT_1a v ... vT~(n_1)a)
n n n
so, lim - log N (T) < h(T,a) = h(T),
Therefore co(T) 2 h(T). //
Consider our examples:
(1) Let A be an expansive automorphism of the torus & .
N (A) = |{x: Anx = x}| = |Kernel (An-I)|
|det ([A]n - I)| (by the proposition below)
I 71" u? - i>|
1
1
whe
re the X. are the eigenvalues of the matrix [A]. So,
cù(A) = lim - £ (log |xn - l|),
n— n X.
l
If |\.| > 1 then
i log |Xn - 1| = £ Clog |X.|n + log |1 - X7n|]
log |Xi| + 0 = log |X£|

180
If |\.I < 1 then
£ log |Xn - 1| - 0.
So, cù(A) = L log |\.| = h(A) by Theorem 6.14.
|X.|>1
Therefore, in this case we have equality.
(2) If T is the two-sided shift on k symbols then
NR(T) = kn and co(T) = log k = h(T),
so that here too we have equality.
In these examples co(T) and h(T) coincide. However, this is
not true for all expansive homeomorphisms as there are examples of
minimal expansive homeomorphisms ( co(T) = 0 since minimality implies
there are no periodic points when the space is infinite) with positive
topological entropy (due to Furstenberg).
Problem:
If T is expansive and has a dense set of periodic points then
is it true that co(T) = h(T)?
Bowen has shown this to be true under the stronger assumption
* * ft
that T is an axiom A homeomorphism.
In example (1) we used the following:
Proposition :
If B: K ■* Kn is an endomorphism of Kn onto Kn with
corresponding matrix [B], (so det [B] i 0) then B is a |det [B]|-to-
one map.
(We used the case B = An - I where [A] has no roots of unity
as eigenvalues.)

181
Proof: [B] is an invertible matrix since det [B] ï 0. Thus,
we can write [B] = E,E„...E where the E. are elementary matrices.
Since [B] has integer entries, the E. have rational entries. Each
E. is one of the following forms:
(1) I with two rows interchanged,
(2) I with one row multiplied by c € Q, or
(3) I with the j-th row replaced by j-th row + c(k-th row), c € Q.
For each E., choose e. € Z such that e.E. has integer entries.
In case (1), e.E. induces an |e.| -to-one map of K , i.e.., an
|e.|n|det E.|-to-one map. In case (2), eiE4 induces an |e.| |c|-to-
one map of Kn, i.e., an |e.|n|det E.|-to-one map. In case (3),
e.E. induces an |e.| |det E.|-to-one map. Let C be the endomor-
phism of Kn determined by e.E.. Let b' be the endomorphism
determined by e....e [B]. Then B' = C, o .. . °C . If B' is a
b'-to-one map and B is a b-to-one map then since B = C, o ... o r
But, also
since [B ] = e, ...e [B], so
r
IT
i=l
r
TT
i=l
r
TT
i=l
b'
|e.|n|det E.|
|e. |n • TT |det E
i=l
|e.|n|det [B]|.
= TT |e.|n-b
i=l x
b = Idet [B]|. //
Consider <b(T) = lim ±. log N (T).
n-*» n n
co(T) is connected with the ^-function which was introduced (for

182
diffeomorphisms) by Artin and Mazur [1]:
If T: X -* X is a homeomorphism such that N (T) < - for all
n > 0 then we set
ÇT(z) - exp(nÇin-znNn(T)),
z € C.
Note that
<b(T) = log
radius of convergence of £T
(Artin and Mazur showed that "most" diffeomorphisms have a ^-function
with positive radius of convergence, and Smale suggested that the
^-function might be a rational function of z for "most"
diffeomorphisms. Manning [1] has shown this to be the case for axiom A
diffeomorphisms but results of Simon [1] have answered Smale's conjecture
3
negatively (for K ).)
It is known that there are no expansive homeomorphisms of S ,
2
but not known if there are any on S . It seems reasonable to ask
whether a compact metric space admitting an expansive homeomorphism
is finite-dimensional.
§6. Examples
We consider the topological entropy of some examples.
(1) Isometries have zero entropy. This is clear from Bowen's
definition. Hence rotations on compact metric groups and all topologi-
cally transitive homeomorphisms with topological discrete spectrum
have zero entropy (Theorem 5.8).
(2) The two-sided shift on k symbols has entropy log k. This is
proved by considering the obvious generator.
(3) Any homeomorphism of K has zero topological entropy.

183
Proof: Let T: K -* K be a homeomorphism. T maps intervals to
intervals as the intervals are the connected subsets. Suppose the
circle has length 1.
Choose e > 0 such that
d(x,y) < e ■=> d(T"1x,T"1y) 2 1/"+.
Consider spanning sets for K with respect to T. Clearly,
r,(e,K) S Cl/e] + 1, where [•] denotes the least integer function.
We estimate r (e,K).
Suppose we have an (n-l,e)-spanning set F of minimal
cardinality r _,(e,K). Consider the points of Tn F and the intervals
they determine. Add points to this set so that the new intervals
have length < e. We have added at most I — + 1 points. Let
F' = F U T"(n-1)(these new points).
We claim that F* is an (n,e)-spanning set for K. Let x € K. Then
3 y € F 3
max d(T1x,T1y) £ e.
0£i£n-2
If d(Tn~ x,Tn~ y) £ e then our claim is proved. If there is no
y € F with both these properties, choose a y € F 3
max d(T1x,T1y) £ e.
0<i£n-2
Consider the interval between T ~ x and T y which is mapped by
T~ to the e-interval [Tn~ x,Tn~ y]. Choose a point Tn~ z, z € F1
inside the chosen interval [Tn~ x,Tn~ y] and with d(Tn x,Tn~ z)se.
Then Tn-2z lies in the e-interval [Tn_2x,Tn-2y] and so,
d(Tn_2z,Tn_2x) s e. The e-interval [Tn_2x,Tn_2y] is mapped by T_1
to an interval of length £ 1/M-, and hence to the e-interval

184
[Tn_3x,Tn_3y]. So, since Tn-3z is in this interval,
d(Tn-3x,Tn~ z) £ e. Similarly, by induction
d(T1x,T1z) £ e V i, 0 £ i £ n-1.
Thus, F is an (n,e)-spanning set for K. So,
r (e,K) S rn_1(e,K) + [1/e] + 1
2 n([l/e] + 1).
Therefore, r_(e,K) = lim -log r (e,K) = 0,
so, h(T) = 0. //
Corollary:
Any homeomorphism of [0,1] has zero topological entropy.
Proof: T: [0,1] -> [0,1] has either T(0) = 0 and T(l) = 1
2
or T(0) = 1 and T(l) = 0. In both cases T induces a
homeomorphism of K. //
(4) If T: Mm -* Mm is a differentiable map of an m-dimensional
Riemannian manifold M111 with Riemannian metric || • || , then
h (T) < max {0 , m log sup ||dT || }
p x€M x
where dT : M ->■ M_, , is the derivative of T at x and p is the
x x T(x)
metric on M determined by the Riemannian metric. This has been
proved by several people.
--2--

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SSSR, vol. 119, pp. 861-864, 1958.
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vol. 149, pp. 453-464, 1970.
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[1] La théorie générale de la mesure dans son application à
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A. G. Kushnirenko:
[1] Metric invariants of entropy type, Uspehi Matematiceskih
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192
D. A. Lind:
[1] Ergodic automorphisms of the infinite torus are Bernoulli,
to appear.
G. W. Mackey:
[1] Ergodic Theory and its significance for statistical
mechanics and probability theory, Advances in Mathematics,
vol. 12, pp. 178-268, 1974.
A. Manning:
[1] Axiom A diffeomorphisms have rational zeta functions,
Bulletin of the London Mathematical Society, vol. 3,
pp. 215-220, 1971.
L. Markus:
[1] Lectures in Differentiable Dynamics, American Mathematical
Society Regional Conference Series, no. 3, 1971.
R. McCabe and P. C. Shields:
[1] A class of Markov shifts which are Bernoulli shifts,
Advances in Mathematics, vol. 6, pp. 323-328, 1971.
L. D. Meshalkin:
[1] A case of isomorphism of Bernoulli schemes, Doklady
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Z. Nitecki:
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D. S. Ornstein:
[1] Bernoulli shifts with the same entropy are isomorphic,
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in Mathematics, vol. 5, pp. 349-364, 1970.
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Society, vol. 77, pp. 878-890, 1971.
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62, 1973.
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Mathematics, vol. 10, pp. 124-142, 1973.
[8] A K-automorphism with no square root and Pinsker's
conjecture, Advances in Mathematics, vol. 10, pp. 89-102, 1973.
[9] A mixing transformation for which Pinsker's conjecture
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[1] An uncountable family of K-automorphisms, Advances in
Mathematics, vol. 10, pp. 63-88, 1973.
[2] Mixing Markov shifts of kernel type are Bernoulli, Advances
in Mathematics, vol. 10, pp. 143-146, 1973.
D. S. Ornstein and B. Weiss:
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194
W. Parry.
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195
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groups, Izvestija Akademii Nauk, Serija Matematiceskaja,
vol. 13, pp. 329-340, 1949 (Russian).
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[1] Statistical mechanics on a compact set with Z action
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196
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[6] Construction of Markov partitioning, Funkcional'nyi
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S. Smale:
[1] Differentiable dynamical systems, Bulletin of the American
Mathematical Society, vol. 73, pp. 747-817, 1967.
M. Smordinsky:
[1] On Ornstein's isomorphism theorem for Bernoulli shifts,
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[2] Ergodic Theory, Entropy, Springer Lecture Notes, no. 214,
1971.
B. Weiss:
[1] The isomorphism problem in ergodic theory, Bulletin of the
American Mathematical Society, vol. 78, pp. 668-684, 1972.

Index
affine transformation 18
aperiodic 89
Bernoulli shift 18, 98
character theory 10
complete invariant 96
completely positive entropy 104-
conditional entropy 7 6
conjugate transformations 53
continuous spectrum 4-6
direct product of probability spaces 5
direct product of transformations 44
discrete spectrum, measure theoretic 64
topological 12 5
distal 12 0
eigenfunction, measure theoretic 45, 63
topological 12 3
eigenvalue, measure theoretic 45
topological 123
endomorphisms of tori 12
entropy 72
entropy of affine transformations of tori 168
of Bernoulli shifts 95
of rotations 93
ergodicity 21
ergodic theorem, Birkoff 29
maximal 32
LP 36
exact endomorphism 111
expansive homeomorphism 171
generator, measure-theoretic 88
topological 168
Haar measure 9
Hahn-Kolmogorov Extension Theorem 4
Hilbert spaces 8
induced operator on Lp 3 2
invariant 5 7
invariant measures for homeomorphisms 128
isomorphism 5 2
Kolmogorov automorphism 101
Kolmogorov - Sinai Theorem 87
Krylov and Bogolioubov Theorem 132

198
Lebesgue Covering Lemma
Lebesgue spectrum
Markov chain
measure algebra
measure preserving transformation
measure theory
minimal homeomorphism
minimal set
mixing
nilmanifold
non-invertible transformation
normal number
orbit
partitions
periodic point
Pinsker algebra
Poincafe Recurrence Theorem
pure point spectrum
recurrence
refinement of open covers
Riesz Representation Theorem
rotations on groups
semi-simple homeomorphism
separated set
sequence entropy
a-algebra
spanning set
spectral isomorphism
spectral theorem
sub-a-algebras
topological conjugacy
topological entropy
connection with measure theoretic entropy
topological entropy of affine
transformations of tori
of Bernoulli shifts
of homeomorphisms of the circle
topological transitivity
uniformly equivalent metrics
uniquely ergodic
151
60
105
53
16
3
113
lit
37
106
110
31
113
70
115
107
20
64
20
140
130
67
115
14-6
108
3
146
54
46
70
122
143, 147
155
168
182
182
117
150
135

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