Chapter I Ergodic Theory

CHAPTER

I Ergodic Theory

1. ABSTRACT DYNAMICAL SYSTEMS

Ergodic theory may be defined to be the study of transformations or groups of transformations, which are defined on some measure space, which are measurable with respect to the measure structure of that space, and which leave invariant the measure of all measurable subsets of the space. In this chapter we shall concern ourselves with the theory of a single measure-preserving transformation and its iterates. This will make it possible to display the essential features of ergodic theory without becoming involved in unnecessary complications of notation and the intricacies of group theory. It should, however, be pointed out that most of the classical applications of ergodic theory require the consideration of a continuous group of transformations. It is customary in ergodic theory to assume that the underlying space is either a finite or a-finite measure space. We shall assume, except in some of the exercises, that the measure is finite and normalized to have total measure one. It is commonly further assumed that the measure space is separable (equivalently, that the space of square-integrable complex-valued functions on this measure space is a separable Hilbert space). We shall not make this assumption, principally because it would rule out some of our most interesting examples and our principal structure theorems in 1

2

I.

ERGODIC THEORY

Chapter III. There seems to be no compelling reason to impose the condition of separability, provided that we do our measure theory, in Chapter IV, for example, in terms of a-algebras rather than partitions. Let X be a nonempty set. Let .a be a a-algebra of subsets of X. In other words, 4 contains the empty set Qr and the set X and is closed under the formation of countable unions, countable intersections, and complements. We make no further assumptions about 4. Let p be a normalized measure on ( X , 93).That is, p is a nonnegative, real-valued, countably additive function defined on a, with p ( X ) = 1. A function 4: X + X is measurable if # - ‘ ( A ) E whenever A E &?. Measurability for a function from one measure space to another is similarly defined. The measurable function #: X 3 X is said to be a measure-preserving transformation if p ( + - ’ ( A ) ) = p ( A ) for all A E &?.It is an invertible measurepreserving transformation if it is one-to-one (monic) and if 4-l is also measurable. In this case 4- is also a measure-preserving transformation. Measure-preserving transformations arise, for example, in the study of classical dynamical systems. In this case 4 is first obtained as a continuous transformation of some (compact) topological space, and the existence of an invariant measure p, that is a measure preserved by #, is proved. The system (X,4, p, 4) is then abstracted from the topological setting. For this reason, we shall refer in this chapter to abstract dynamical systems.

Definition 1.1 An abstract dynamical system is a quadruple = ( X , a, 4), where X is a nonempty set, W is a a-algebra of subsets of X, p is a normalized measure defined on 4, and # is a measure-preserving transformation of X.We shall say that is inoertible if 4 is invertible. p,

While our principal object of study is, of course, the transformation 4, we adopt the above notation and terminology for a variety of reasons. For example, we shall have occasion to consider as different dynamical systems two quadruples @ = ( X , A?,p, 4) and @’ = (X,W‘,p, #), which differ only in the class of measurable sets. In order to avoid some trivialities as well as some embarrassing technical difficulties of measure theory, we shall adopt the following notion of equivalence of abstract dynamical systems.

Definition 1.2 The dynamical systems @ = (X, 93, p, 4) and 4’) are equivalenr if there exists a mapping +*: # 3 28 which is monic and epic, and which satisfies @’ = (X’, w’,p’,

p(+*(B’)) = p’(B’)

(B’E a )

1.

3

ABSTRACT DYNAMICAL SYSTEMS

and p [ d - l ( $ * ( B ’ ) ) A $*(r#-’(H))]= 0

(B’ E

a‘).

Here we have used the symbol A to denote the symmerric difference CAD=(C-D)u(D-C) of the sets C and D. Of course, if $: X + X ’ is an invertible measure-preserving transformation such that $4 = 4’1) modulo sets of measure zero, then its adjoint $* defined by $*(B) = $- ‘ ( B ) satisfies the above requirements, so that Q, and W are equivalent. However, it is not always possible to find such a $ for equivalent systems Q, and 0’. Let us consider some examples of abstract dynamical systems. Let X = [0, 11 be the set of real numbers between 0 and a denote the Borel subsets of X and p the restriction of Lebesgue measure to X . Define $(x) for each x E X to be the fractional part of .Y + a, where 0 < a < 1. It is easily verified that 4 preserves the length of any interval and hence (see Exercise 1) the measure of any set in A?. Let X’ = K = { z : ( z I = 1) be the set of complex numbers of absolute value one. Let J‘ denote the Borel subsets of X ’ and p’ the normalized linear measure on X’. Define 4’: X ’ + X ’ by 4(z) = e’”‘z. Then 4f= ( X ’ , a’,p’, 4’) is a dynamical system, which is easily seen to be a,p, 4). equivalent to 0 = (X, Let X” = X x X be the unit square, and define tff”(x, y) = (4(x), y). If p” is two-dimensional Lebesgue measure, then W = ( X ” , X’,p”, 4”) will but not equivalent be equivalent to 0 provided that X’ = ( A x X : A E a}, if L9’’is taken to be all the Borel subsets of X “ . Example I

1 inclusive. Let

Example 2 Let X , 2, and ,u be as in the previous example, and define 4(x) to be the fractional part of kx, where k is a positive integer. Equivalently, define 4’ on X’ by &’(z)= zk. This is an example of a noninvertible dynamical system (for k > 1). Since 4([0, l l k ] ) = X , it is not generally true for noninvertible systems that p ( 4 ( A ) )= p ( A ) , even when $ ( A ) is measurable. Example 3

(Shift transformations) Let X,

= (0,

be a finite set of k points. Let

1, ..., k - 1).

a,,denote

the class of all subsets of X , ,

4

I.

ERGODIC THEORY

and let p n be the measure obtained by assigning to the point j the mass Pnj 1

= { P n O 9 P n l ? * * . ? Pn, k-

C(n

1).

We can form a measure space (X,93, p) by taking the infinite product m

(X,a,

=

X

n=O

( X n , an,Pn)

or the two-sided product m

(X‘, P’)= X ( X n T a n , pn)* g

y

n=-m

That is, X or X’consists of (one- or two-sided) sequences of elements of X,,93 or 9 is the smallest a-algebra containing all “cylinder sets”

c = {x x : (Xnir .. .,xn,)

A) =

u n I

(s,, ..., S,)E A j = l

{x : x n j = sj),

and p or p’ assigns to C the measure

Now define 4 and @ on X and X’,respectively, by 4(x) = y or @(x) = y, where y n = xn+ (all n). Noting that

4-YC)=

u n 4-yY: u n iX + I

(Si,

.. . , 7,)

E

A

J=

Ynj = s , ~

1

I

= (Si’

...

1

S,)E

Aj= 1

: xfl,

=sJ~,

we see that 4-l (or +’-’) carries cylinder sets into cylinder sets, hence is measurable. Clearly, it will be measure-preserving iff P“j

= pj

( j = 0,1, . .., k - I),

independently of n. Note that 4’ is invertible, but that 4 is a k-to-one transformation. In fact, 0 is equivalent to the system 0 of Example 2. To see this, we need only express each x E [0, 13 in its k-adic expansion, thus obtaining an almost one-to-one correspondence between it and the one-sided sequence space X.

1. ABSTRACT DYNAMICAL SYSTEMS

5

If CP = (X, a,p, 4) is an abstract dynamical system, then 4 determines a transformation Tb of (real- or complex-valued) functions on X, defined by the formula T , ( f ( x ) )= f ( # ( x ) ) .Iffis measurable with respect to W,then so is T , f . If f is integrable, then T,f is integrable, and jT,f d p = J f d p . This follows for simple functions from the fact that 4 is measure preserving and in the general case by a limiting argument. Recall that L, = L p ( p )= L,(X, B, p) (1 I p < 00) denotes the set of all measurable real- or complex-valued functions f defined on X for which j l f ( x ) l p p ( d x )< 00, and that L, with the norm

is a complete, normed linear space. We denote, as usual, by L , the space of p-essentially bounded functions with the p-essential supremum norm. In this chapter we shall be chiefly concerned with real L,; that is, the functions in L, will be assumed to be real valued. Note, however, that the ergodic theorems of Section 2 are valid in complex L,, and that the spectral theory introduced in Section 3 requires consideration of complex L,. Since T, 1 f 1 = 1 T,f I, it is clear that the linear transformation T+ maps L, into L, for each p , 1 Ip I co, and that IIT,fl(, = Ilfll,. That is, Tb is an isometry on L,. If is an invertible measure-preserving transformation, then T, is an invertible isometry, with T i = Tb-I . In the case p = 2, this means that T, is unitary. We define a doubly stochastic operator on L, (1 Ip I co), so called because of its origins in probability theory and its analogy to doubly stochastic matrices, as follows.

Definition 1.3 A linear operator T defined for functions on X is doubly stochastic if it maps L , into L , and satisfies for all f E L , the following conditions: 1. f 2 0 a T f 2 0 ; 2. j x Tf dP = Jx f d p ; 3. TI = 1.

Here we use the symbols L and = in the p-almost-everywhere sense, and we denote by 1 the function whose constant value is 1. For each continuous linear operator T on L,, there is a well-defined continuous linear operator T*, called the adjoint of T , defined on L,, where 1 I p < 00 and I/p + l/q = 1 ( q = 00 for p = 1). They are related by

6

I.

ERGODIC THEORY

(Tf, g) = (f, T*g) for all f~ L,, gE L,. The symbol “inner product”

(JT

g) denotes the

where C is the complex conjugate of c and the bar may be ignored for real L, .

Proposition 1.1 If T is a doubly stochastic operator, then T maps L, into L,for each p , 1 I p I co, with (IT([, I 1 and llTlll = llTllm= 1. Moreover, T* is also doubly stochastic. Proof We show first that IITI(, = 1. This makes sense because L, G L,, and T is defined on L,. For f E L,, let f and f - denote, respectively, the positive and negative parts of f . Thus f = J’ - f - , I f 1 = f + + f - , and f +, f - 2 0. By property 1 of Definition 1.3, Tf 2 0 and Tf - 2 0. Thus +

+

lTfl = IT(f+ -

f-)I

= IT’+

- Tf-I I Tf’ + Tf-

= Tlfl.

(1)

By properties 1 and 3, Combining inequalities (1) and (2) yields ( 1 Tf(l,I 1 f l l , , or ( 1 TI(,I 1. Since T1 = 1, it follows that IJTIJ, = 1. Note that we used only properties 1 and 3 and the linearity of T to show that liTIJ, = 1. Now property 2 is equivalent to saying that (Tf,1) = (f, T*l) = (f, 1) for all f E L,. That is, property 2 is equivalent to T*l = 1. It follows as above that JIT*(l,= IJTIJ,= 1. The inequality )(TllpI.1 for 1 < p < 00 now follows as an application of the Riesz convexity theorem [16, p. 5261. (See also Exercise 7.) To complete the proof we need to show that T* maps L, into L, and satisfies condition 2. Iff€ L , , then J T*fdp = (1, T*f) = (T1,f) = (1,f) = Jf dp. Since L, is dense in L,, it follows that T* has a unique continuous extension to all of L, satisfying f T*f dp = Jf dp for all f E L,. Integrating inequality (1) above with T replaced by T* shows that T* maps L, into L,. 1 It is interesting to note that the properties mentioned earlier for doubly stochastic operators arising from dynamical systems characterize these operators. Specifically, we have the following.

2.

ERGODIC THEOREMS

7

Proposition 1.2 Suppose that ( X , 93, p) is the unit interval with Lebesgue measure on the Borel sets. Then the operators of the form for some dynamical system (4 = ( X , 93, p, 4) are just exactly the doubly stochastic operators which are isometries on L , ( X , B, p). Moreover, (4 is is unitary. invertible iff We shall not have occasion to use this result, and so will not give the proof. See, however, Exercise 6 at the end of this chapter.

2.

ERGODIC THEOREMS

In the previous section we said that ergodic theory might be defined as the study of measure-preserving transformations. A more restrictive definition would be the study of the asymptotic behavior of the iterates c#P of such a transformation. Indeed, the historical beginning of this discipline might be placed at the proof by G. D. Birkhoff in 1931 of the so-called individual ergodic theorem (Theorem 1.2) or the earlier proof by H. Poincare in 1912 of the recurrence theorem (Theorem 1.5). In this section we shall look at these theorems as well as several others of a similar nature. We shall refer to these theorems collectively as ergodic theorems. Some of them involve the iterates 4" of a measure-preserving transformation, while others involve the iterates T" of an operator having some or all of the properties of Definition 1.3. We make no pretense at completeness or ultimate generality in our selection of ergodic theorems, but give only a representative sample of those we believe have had the most impact on ergodic theory and its applications. One further historical note seems to be in order at this point. In retrospect it is clear that the mean and individual ergodic theorems for a measure-preserving transformation were anticipated considerably earlier by the (Weak) Law of Large Numbers of J. Bernoulli (1713) and the Strong Law of Large Numbers of E. Borel (1909) for Bernoulli sequences of random variables. The identification of these latter theorems as ergodic theorems only awaited the invention of measure theory by Borel and Lebesgue and its application by A. Kolmogorov in 1933 to the foundations of probability. We begin with one of the so-called maximal ergodic theorems, this one due to E. Hopf. As before, let ( X , 93, p) be a normalized measure space. Let T be an operator on L , = L , ( X , B, p). We need only assume that T has property 1 of Definition 1.3 and a weakened form of 2,

8

I.

ERGODIC THEORY

namely 1 TI1 I1. Such an operator is called a contraction. We introduce the following notation:

T,fW =

c TWx),

n- 1

k=O

B * ( f ) = {x : sup T, f ( x ) > 0},

B n * ( f )= {x : max

I
n

where f

E

q f ( x ) > 0},

L,.

Theorem 1.1 (Hopfmaximal ergodic theorem) For each f

Proof

E Ll

(A. Garsia [24]) Let (n = 1, 2, ...).

fn(x)= max q f ( x ) Isksn

Thenf = fl I f2 I. - .and Bn* = B n * ( f )= {x : f , ( x ) > 0) is an increasing sequence of sets with union B * ( f ) = B*. Also, since T 2 0,

+

Tlf = f If Tf,+ q+lf= f + T ( T , f )2 f

+ Tf,+

(1 I k In)

so that

f n 5 fn+1

2

j"

X

fn+

I

f + Tfn+ d~

-5

X

Un+

d~ =

IIfn+II1

- I I T f n + I I 1 2 0.

The last inequality comes from the assumption that IITI(, I 1. Letting n -,co, we obtain the desired result

[,*f dP 2 0. 1 Now let us introduce the further notation:

I

I

1 A * ( f ; a ) = x : sup - T n f ( x )> a , n

n

2.

9

ERGODIC THEOREMS

Corollary 1.1.1 For each f

E

L , and each real a and

p we have

and

Proof To prove (4) apply (3) to the function h = f - a E L , and observe that T, h ( x ) > 0 iff (l/n)T,,f ( x ) > a. Inequality ( 5 ) then follows by applying (4) to g = -f and taking a = -pa I induced by the After particularizing this corollary to the operator dynamical system a, it is a relatively easy task to prove the most celebrated of the ergodic theorems, that of Birkhoff.

Theorem 1.2 (Birkhoflindividual ergodic theorem) Let = ( X , &?, p, (6) be an abstract dynamical system, and suppose that f E L , = L , ( X , 93,/A). Then there exists a functionf E L , such that i

Proof

n-1

(F. Riesz) Let us denote f ( x ) = lim sup n-m

f * ( x ) = sup n

1 ~

n

T, f ( x ) ,

n1 T,f ( x ) ,

1 f ( x ) = lim inf - T,,f ( x ) n+m n

1 f * ( x ) = inf T, f ( x ) , ~

n "

so that in Corollary 1.1.1

A*($; a ) = {x : f * ( x ) > a),

For fixed a and

p with p <

AJf;

8) = { x : f J x ) < Bl.

a, let

A(a, p) = {x : f ( x ) < p < a

-= J ( x ) } .

(7 )

Since f ( ( 6 ( x ) )= f ( x ) and f ( 4 ( x ) )= f ( x ) for all x E X , it is clear that A(a, p)+A(a, 8). Assuming that p(A(a, B)) = y > 0, we can apply

4:

10

ERGODIC THEORY

I.

Corollary 1.1.1 to the dynamical system = (A(& p), B n A(a, B), (l/y)p, 4). Since f* I f r f ~ f * we , have for O m , pthat A * ( f ; a ) = A J f ; p) = A(a, 8). It follows that 1

al-

I,,,

ps which contradicts p < a. Thus we have p(A(a, p)) = 0. Since Y

A = {x : f ( x )

p!

dp I

u

-= f ( x ) } = ry

A(a, B),

pta; fl rational

it follows that p ( A ) = 0. Thus f ( x ) = f ( x ) p-a.e., and the proof of convergence is complete. To see that f~ L,, note that

By Fatou’s lemma,

Remarks Z Much has been done in the way of proving individual (that is, pointwise convergence) ergodic theorems for operators. See, for example, the excellent account by Garsia in [24]. A direct generalization of Theorem 1.2 to doubly stochastic operators yields the Hopf ergodic theorem. The proof again is based on Theorem 1.1. The same result with weaker hypotheses was proved by Dunford and Schwartz (see [ 161 or [24]). Recently, using the notion of “dilation of an operator,” Akcoglu [3] has proved pointwise convergence for (positive) contractions on L,, 1 < p < 00. 2 In the case of a discrete (completely atomic) measure space, a classical theorem of Kolmogorov yields convergence as in Eq. (6) for operators T only assumed to satisfy properties 1 and 3 of Definition 1.3. This theorem, usually stated in terms of convergence of a sequence of matrices, is basic in the analysis of finite or denumerable Markov chains. 3 Many of these theorems, including Theorem 1.2, are also valid when p is a o-finite measure. However, the limit function f may be uninteresting in this case (Exercise 9). A more sophisticated result, which also includes almost all of the theorems mentioned so far, is the following.

2.

11

ERGODIC THEOREMS

Theorem 1.3 (R. Chacon-D. Ornstein) Suppose ( X , 99, p) is a j n i t e or a-jinite measure space and T is a linear operator on Ll = L l ( X , 94, p) satisfying (i) T 2 0 and (ii) I(T(1,I1. Then for each f , g E L, with g 2 0, the limit

exists and i s j n i t e almost everywhere that

SUP, T , g ( x ) > 0.

The proof of this theorem is complicated and will not be given here (see [24, p. 30 IT.]). Instead, we proceed now to a fairly general “mean ergodic theorem,” that is, one asserting convergence in L, . If Q, is a dynamical system, we shall see that for each f e L, the sequence ( l / n ) T ,f converges in the norm topology of L,Jl Ip < a).It follows (Exercise 1 1 ) that the limit must coincide with f almost everywhete. Thus

and, in particular, f E L,. In the following, we assume as before that ( X , 3, p) is a normalized, finite measure space. Theorem 1.4 (Yosida mean ergodic theorem) Let T be a doubly stochastic operator and f E L, . Then there exists f * E L, such that Tkf-f* IIP

=o.

(8)

Proof Suppose h is a function on X with T h Ih. Then g = h - T h 2 0 and J x g dp = 0. It follows that g = 0; that is, Th = h. According to Proposition 1.1, the same is true with T replaced by its adjoint T * . In particular, if T*h, = h, and T*h, = h,, then by the positivity of T * and the previous remark, T*(hl A h,) = h, A h,, where h, A h, denotes the infimum of h, and h,, defined by (hi

A

h,)(x) = min(h,(x), h,(X)).

Suppose f = g - Tg with g E L,. Then T,f = g

-

T”g. It follows that

12

I.

ERGODIC THEORY

as n-t co. Thus (8) holds with f* = 0 for all f in the subspace %, = { g - Tg : g E L,}. Likewise, (8) holds for all elements of X, = {f E L, : Tf = f},withf* = f . We shall show that X, + 3, is dense in L,. It will follow that (8) is valid for all f E L,. For if f in L, and (8) is valid for each fk, then

which implies that (8) holds also for f . To prove that %, + 3, is dense in L,, we shall show that the only F E L, = L,* (l/p + l / q = 1 for p > 1, and q = c13 for p = l), which is orthogonal to both X,and X 2 ,is the zero function. Suppose then that F is such a function. It follows that ( F , g - Tg) = ( F , g ) - (T*F, g ) = 0 for all g E L,, and hence that T*F = F . Let c be a fixed real number, and set A = { x : F ( x ) > c}. For each E > 0 we define

+

g, = (1/E)[(c &)

A

F - (C

A

F)].

Then 0 I g , I 1 and g, 7 x A , the characteristic function of the set A, as E 10. It follows by the monotone convergence theorem that T*g, 1 T*xA (Exercise 8). On the other hand,

+ E)

T*g, = (l/e)[T*((c = ( I / E ) [ ( C -k

since F , c, and c

E) A

F ) - T*(c A F ) ] F - (C A F ) ] = g c , A

+ E are all invariant functions for T*. Thus T * z A= lim T*g, = lim g , = z A . e-0

(9)

C-0

We shall show, in fact, TxA = x A . If B E 28 is arbitrary, then by (9) and the positivity of T T*xA n B 5 T*xA = X A

so that

and

T*xA

I T*xB,

2.

13

ERGODIC THEOREMS

Likewise, or

Since B was arbitrary, it follows that T z A= x A , as asserted. Now x A E L , E L,, and hence x A E 3 , . Since F is assumed orthogonal to S,,this means that (XA, F) =

1

*

[F>c]

F dp = 0.

Since this is true for all real c, we must have F ( x ) = 0 a.e.

I

We conclude this section with a third type of ergodic theorem, the recurrence theorem of Poincare. p, 4) be an abstract dynamical Theorem 1.5 (Poincare) Let 0 = ( X , 9, system, and let A E 9. Then for almost every x E A there is a positive integer n = nA(x)such that @ ( x ) E A .

Proof Let B=A-

m

a3

n= 1

n=l

( J { x E A : @ " ' x ) E A } =n ( A - $ - " ( A ) ) .

Since

+-"(B) =

n m

n= 1

(4-m(~)

- 4-("+")(~)),

it isclear that the sets B, 4- ' ( B ) ,$-'(B), . . . are pairwise disjoint, measurable sets. Since p ( 4 - " ( B ) )= p ( B ) for each n, and since p ( X ) = 1, it follows that P(B) = 0. I

14

I.

ERGODIC THEORY

3. ERGODICITY AND

MIXING

So far we know very little about the limit function f in Theorem 1.2. We know (see Exercise 11) that it coincides a.e. with the function f * of Theorem 1.4, and that J f d p = J f dp. Of course, in some special cases we can completely identifyf. For example, iffis an invariant function, T f = f , then f = f. We know that when f = g - T g for some g E L,, we have f = 0. There is one more situation in which we can completely identify f. This is when the dynamical system # is ergodic. Definition 1.4 The abstract dynamical system # = ( X , 93, p, 4) is ergodic if 4 - ' ( A ) = A, A E 93, implies either p(A) = 0 or p ( a ) = 0. A doubly stochastic operator T is ergodic if T f = f , f E L,, implies that f is essentially a constant function, that is, f ( x ) = c a.e. Proposition 1.3 Zf @ is ergodic, then the induced operator T4 is ergodic. Proof Suppose T+f = f . For each positive integer n and each integer k let

Since 4 is ergodic and 4 - ' ( X ( k , n)) = X ( k , n), it follows that p ( X ( k , n ) ) = 0 or p ( X ( k , n ) ) = 1. For each n there must be exactly one k with p ( X ( k , n)) = 1. Denote it by k(n). It follows that X , = X(k(n), n) has measure 1. Clearly then, there exists a constant c such that x , = {x : f ( x ) = c}.

n."=o

Proposition 1.4 Zf# is ergodic, then

lim

n-ql

1 n-1 -

C f(@(x))=

k-0

5 f dp

a.e.

X

for each f E L,, and

for each A, B E 93. Conversely, if (12) holds for all for all A, B E 9?,then 4 is ergodic.

f E L,, or if (13) holds

3. ERGODICITY

AND MIXING

15

Proof The validity of (12) follows from T , f = f and f dp = f dp. If we set f = ,yg, then f ( 4 k ( x ) )= ~ + - y ~ ) ( xIntegrating ). (12) over A and applying the bounded convergence theorem yields (13). Since (12) implies (13), it only remains to show that the validity of (13) for all A, B E 93 implies that @ is ergodic. Suppose 4- ' ( B ) = B and set A = 4. The left side of (16) is zero then, and so either p ( B ) = 0 or p(B) = 0. 1

The equality (12) is very closely related to the origins of ergodic theory in statistical mechanics. If we think of the sequence &"x) as unfolding in time, then (12) is a statement of the ergodic hypothesis, namely, that time averages (of integrable functions) coincide with space (or phase) averages. In probability theory, (12) provides the foundation for a method of estimating parameters for (ergodic) stationary processes. The significance of equality (13) is related to the recurrence theorem of Poincare (Theorem 1.5). The latter theorem implies that, for a set A of positive measure, almost every point of A returns to A infinitely often. It gives us no information, however, as to how many points of A return to A at the nth step of the process, or, more generally, how many points of A are in the measurable set B after n steps. The proper measure of this number is p ( A n 4-"(B)).Equality (13) tells us that asymptotically this number is on the average for different values of n proportional to the sizes of A and B. It may in fact (Exercise 18) never, for a given value of n, be close to p ( A ) p ( B ) . Intuition tell us that for certain processes we should, in fact, have p ( A n 4-"(B))converging to p ( A ) p ( B ) as n -,co.When this is true, the process is said to be mixing (or strongly mixing).

Definition 1.5 The dynamical system lim p ( A n 4 - " ( B ) )= p ( A ) p ( B )

is (strongly) mixing if for all A, B E 93.

n-r m

To borrow an illustrative example from Halmos [32], suppose that a mixture is made containing 90% gin and 10% vermouth. If the process of stirring the mixture is ergodic, then after sufficient stirring any portion of the container will contain on the average (with respect to the number of stirrings) about 10% vermouth. If the process is a mixing one, the amount of vermouth in the given portion will become and remain close to 10%. Since molecular theory allows for occasional "accidents," such as the kitchen table that rises into the air because all of its molecules are moving in the same direction, we may want to consider a slight weakening of the

16

I.

ERGODIC THEORY

notion of mixing, namely that after a large number of stirrings the amount of vermouth in the distinguished portion of the container will be close to 10 % except for rare occasions. We shall say that a set J of positive integers has density zero if the number of elements in J n { 1, 2, ..., n} divided by n tends to 0 as n -+ 00.

Definition 1.6 The dynamical system @ is weakly mixing if for each A,BE@ lim

n-ra. n4 J

p ( A n +-“(B))= p ( A ) p ( B ) ,

(14)

where J is a set of density zero, which may vary for different choices of A and B. The following proposition shows that weak mixing lies logically between mixing and ergodicity. Proposition 1.5 Let are equivalent:

be an abstract dynamical system. Then the following

(i) @ is weakly mixing; 1 n-1

(ii) lim

-

(iii) lim

-

n-rm

1 Jp(An $ J - ~ ( B-) )p ( A ) p(B)I = 0 ( A , B E a ) ;

k=O

1n-1 n-rm

1 [ p ( A n I # I - ~ ( B ) =) ] p(A)’ ~ p(B)’

k=O

( A , B E a);

(iv) the dynamical system 0’= ( X x X, 39 x ergodic, where (4 x 4 ) b ,Y ) = (4(x), 4 ~ ) ) ; (v)

(P’

a, p

x p,

I#I

x

4)

is

is weakly mixing.

Proof For a bounded sequence {a,,} let us write a = *-limn+ma, provided that a = 1imndm,n 6 J a,, where J has density zero. Then, in general, a = *-lima, n-x

iff lim n+m

1n-1 -

1 )ak- a /

k=O

=O.

(15)

For suppose that a = *-limn-.ma,, with the exceptional set being J . If la,l I b for all n, then

3.

17

ERGODICITY AND MIXING

where (J,I is the number of elements in J n (0, 1, ..., n - 1). Thus limn+ (l/n) CE, 1 ak - a 1 = 0. a, iff J ( E )= To prove the converse, note that a = { n : la - a, 1 L E } has density zero for each E > 0. For if the latter holds, then there exists an increasing sequence of integers n, (m = 1, 2, . . .) such that

A

n 2 n,

I J , ( l / m ) l < n/m.

Setting

we have for each m and

Now suppose that a # *-limn-toaa,. Then there exist such that I J , , ( E ~2) ~nE2 for all n. It follows that

E~

and hence

This completes the proof of equivalence (15). Clearly, then (i) and (ii) are equivalent. Also *-lim p ( A n c#-"(B))= p ( A ) p ( B ) n- m

iff *-lim Ip(A n # - " ( B ) ) - p ( A ) p(B)I

=0

n-. m

iff

*-Iim ( p ( n~ c # - " ( ~ ) ) - p ( ~~ )( B ) J=*O n-+ m

iff

> 0 and

E~

>0

18

I.

ERGODIC THEORY

Now if # is ergodic, then

f kc= O 1/44 4-"(B))- P(A) @)I 1

lim

n-m

n-1

c [P(A

1n-1 =

lim n

n+m

k=O

n 4 - n ( B ) ) Z- 2P(A) P(B) P(A n 4 - n ( q 1

+ C1(42 C1m2 1n - 1

= lim n+m

k=O

p ( A n 4-"(B))' - &Ip(B)', )'

so that in this case (ii) and (iii) are equivalent. However, either (ii) or (iii) implies that # is ergodic. Thus (ii) and (iii) are equivalent. To show that (14) holds for all A, B E g x 9.3, with 4 replaced by 4 x 4, it is sufficient to show that it holds for measurable rectangles. Condition (14) then becomes * - h P ( A n 4-"(B))P(C n 4-n(D))= P L ( 4 n-r m

A C ) P(D)

(A, B, C, D E g). (17)

Since the union of two sets of density zero has density zero, (17) follows from (14). That is, (i) implies (v). Since (iv) obviously follows from (v), it only remains to show that (iv) implies (iii). If (0' is ergodic and A, B E 94,then

cp(A

1 n-1

lim -

n-m

k=O

as was to be shown.

n 4-k(B))2

I

It is time now to discuss some of the spectral properties of the operator on L,. For this purpose, we consider to be operating on complex L, . Iff E L, is a nonzero, complex-valued function such that Tf = 1f for some complex number 1,we say that 1 is an eigenualue and f an eigenfunction of T. The collection of all eigenvalues of T is called the point spectrum of T. If X is a T-invariant subspace (TS= S)of L , containing no eigenfunctions of T, we say that T has continuous spectrum on S.An

19

3. ERGODICITY AND MIXING

eigenvalue I of T is simple if T f = If, Tg = Ig implies that g is a constant multiple of f. If the invariant subspace X has a basis consisting of functions f;:I ( i = 1, 2, 3, . .. ;j = 0, k 1, _+2, . ..) with TAj = fr,j+l for each i and j, we say that T has countable Lebesgue spectrum on X .

Theorem 1.6 Let

Q,

be an invertible abstract dynamical system, and let

q be the induced operator on (complex) L, . Then 1 is an eigenvalue of q,

and all eigenvalues have absolute value 1.

(i) If cf, is ergodic, then all eigenvalues are simple, and they form a subgroup of the multiplicative group K = { z : 1. = I}. (ii) If cf, is weakly mixing, then Tb has continuous spectrum on the complement of the space of constant functions. (iii) I f q has countable Lebesgue spectrum on the complement of the space of constant functions, then Q, is strongly mixing. Constants are eigenfunctions corresponding to the eigenvalue 1. Thus T+ cannot have continuous spectrum or Lebesgue spectrum on any space containing constants. By the complement of the space of constant functions, we mean the uniquely defined space X C L , such that L , = X + {constants} and every function in X is orthogonal to 1, that is, Remarks Z

X={f-j

X

fdp:f~Li).

2 The condition in (ii) is necessary and sufficient for mixing. For a proof see, for example, [32, pp. 39q.

Q,

to be weakly

Proof We have already remarked that the constant functions are invariant, and hence that 1 belongs to the point spectrum. Since q is unitary, all of its eigenvalues have absolute value one. Alternatively, if q f = Al; then T,lf I = 111( f1, and since q1.f I d~ = 121 (fld p = If I d p # 0, it follows that ) I \ = 1.

1

1

1

(i) According to Definition 1.4 and Proposition 1.3, Q, is ergodic iff 1 is a simple eigenvalue for T+. If Q, is ergodic, and if T+f = I f, then T+I f I = so that 1 f 1 must be a constant. If, in addition, q g = Ig, then T+(f / g ) = (f/ g ) , so that f / g is a constant. Finally, if T+f = I If and T+g = I , g, then T + ( f / g )= (,&/I,)(f /g), so that 11/12is an eigenvalue. (ii) Suppose 0 is weakly mixing. Then 0, is ergodic. Suppose Tb f = I f,and let g(x, y) = f ( x ) f(y).Then

If/,

rn)

s ( 4 ( x k 4(YN = f (4b))

=W x )

5) = g(x9 Y ) .

20

ERGODIC THEORY

I.

Thus g must be a constant. Hence f is a constant and I = 1. (iii) If f is a constant, then (T@"f, g) = (f,g) = (f,1)(1, g) for all n and each g E L,. I f f = J j , g = fp4, then (5%g) = (A,j + n r f,) = 0 for all sufficiently large n. Since the functions J J plus the constant function 1 form a basis for L,, it follows that lim (T@Y,9 ) = ( f 7 1)(L 9 )

( f 7

n-r m

9 E L2).

In particular, this is true when f and g are characteristic functions and so CP is strongly mixing. I Let us look again at the examples of Section 1. Example 2 Suppose first that a is irrational. In this case, CP is ergodic but not weakly mixing, hence not strongly mixing. Using the alternate description of Q, on K = { z : IzI = 11, we see that

T@f ( z ) = f(e2aiaz). If f,(z) = z", then T4"f"fZ)= e2nina z n

- n - cf,(z).

Thus f, is an eigenfunction with eigenvalue I = c" # 1. According to Theorem 1.6, Q, is not weakly mixing. On the other hand, any function f E L, can be expanded in a Fourier series:

f

f

anfn?

n=--00

which converges to f in L,. Thus if T# f the expansion

=

X

it follows that f also has

U

T4.f

n=

1-

anc"fn.

30

By the uniqueness of the Fourier coefficients, it follows that a, = a,c" for each n. This means that a, = 0 for n # 0, and s o f i s a constant. If a is rational, then c" = 1 for some positive integer n. Thus T@has nonconstant eigenfunctions, and so 0 is not ergodic. Example 2 This system is equivalent to the system Q, of Example 3. This equivalence clearly preserves all properties of ergodicity and mixing. Note also that there is an induced unitary equivalence of the corresponding L , spaces. The system @ is not invertible, so Theorem 1.6 does not apply.

4.

PRODUCTS AND FACTORS

21

Example 3 0’ is strongly mixing. This follows from Theorem 1.6 by taking fP4 ( x ) =

e2niP.xq/k.

(18)

This example is a special case of a theorem about automorphisms of groups to be proved in Chapter 111. It is fairly clear, and it will follow from a theorem on inverse limits, that the strong mixing property for 0 is equivalent to the same property for 0’. We have not shown yet that there exist systems 0 that are weakly mixing but not strongly mixing. This is a surprisingly difficult task, especially since it is now known that “most” systems are of this type. An example is given at the end of Section 6 and in the exercises.

4.

PRODUCTS AND FACTORS

We begin now the study of methods for constructing new dynamical systems from given ones. This will lead in later chapters to representation theorems, whereby we express more complicated systems in terms of simpler, more familiar ones. The first such construction is the direct product 0 0 R of dynamical systems 0 and R. We have already used a special case of this construction, namely 0 0 0 = B2, in Proposition 1.5. Definition 1.7 We define the direct product m1 @a2of abstract dynamical systems Qi = (Xi, B i , p i , 4i) ( i = 1, 2) by 0 1 @ 0 2 = ( xx1x

, , 2 1 X.ZB,,Pl XP2’41 x 4 2 ) ,

where (41x 4 2 ) ( x 1 , x 2 ) = ( 4 1 ( x 1 ) , 4 2 ( x 2 ) ) . More generally, if 0, = (X,, a,, p,, 4=) is an abstract dynamical system for each 01 E J, we 0, by taking the product measure define the direct product 0 = gmEJ X, and defining structure on the product space X = X, 4 ( x ) = y,

where Y , = 4 A x d

(19)

We shall make use of customary modifications of this notation, such as 0 O2 0 ... 0 0”and @,“=l 0”. Proposition 1.6 The product of a weakly mixing system and an ergodic system is ergodic. The product of two weakly (strongly) mixing systems is weakly (strongly) mixing.

22

I.

ERGODIC THEORY

Proof Let CPl and (D2 be the two systems. It suffices to prove that (13) or (14) or the defining relation for strong mixing holds for pairs of measurable rectangles. That is, we need to show for all A, B E 93, and C, D E B2 that

1 n-1 lim

n+m

-

c Pl(A

k=O

n 4 ; k ( B))PZ(C f-4ik(D)) l

or

or lim Pl(A n 4

a+

00

m ) )P2(C n 4 i n w= P l ( 4 P A B ) PZW) P2(D),

(22)

where CP1 is weakly mixing and O2 is ergodic for (20), both are weakly mixing for (21), and both are strongly mixing for (22). The last one is completely obvious, while (21) depends only on knowing that the union of two sets of density zero has density zero. To prove (20) we note that, for a given E > 0 and for all k larger than some no = no(&), we may replace pl(A n 4Lk(B))by p , ( A ) p , ( B ) + ek with < E, except when k belongs to some set J of density zero. Thus

i

n-1

I

5 no/n + ((n - nO)/n)&+ (l/n)lJnI.

The first and last terms on the right tend to zero as n -,co. Thus the two terms on the left have the same limit, namely the right side of (20). I We shall see in the next section that the product of any number of weakly (strongly) mixing systems is weakly (strongly) mixing. Note, however, that the union of countably many sets of density zero need not have density zero. Suppose that CP = CP, 0 CPz. Define +: X + X, by +(x, y ) = x. It follows easily that $4 = 4,+. That is, the diagram

4.

PRODUCTS AND FACTORS

!*

Xl

dl

'

23

!*

x,

commutes. Moreover, $ is measure-preserving. It is possible to have systems O and (9, related by a map +: X -,XI for which diagram (23) commutes without (9 being of the form O1@ 0 2 . Definition 1.8 We shall say that the dynamical system (9' = ( X l , a,, p,, 4,) is afactor of the system = (X,B, p, 4) if there exists a measure-preserving map $: X + X , such that diagram (23) commutes. In this case, we write Q,, I@ and t,b: (9 -+ O , or O 3 Ol. The map $ is called a homomorphisni of Q, onto a1. If O = 0 02,Q1 is called a direct factor of O. Note As usual in this chapter, when we write $4 = 4,t+h,or indicate it by a diagram like (23), we mean that equality holds pointwise almost everywhere. Suppose that O110.Let g1' = { $ - ' ( A ) : A E B,}. Then Bl' C 9,and = ( X , B,', ,u, 4) are according to Definition 1.2, the systems O, and 0,' equivalent. Thus we may always assume that the factors of O are of this latter form. That is, the factors (D1 of @ may be identified with the sub-8-algebras a, of 29 which are invariant, in the sense that 4-'(Bl) = { 4 - ' ( B ) :B E Bpi> c 9'.Note that the factor Olis an invertible system iff Bl is totally invariant, that is, 4-'(B1)= al (Exercise 22). It might be imagined that two dynamical systems 0,and O2 for which O1 and O210,are isomorphic, hence equivalent in the sense of Definition 1.2. However, it is not known if this is true even in the case where 41and 42 are the identity; that is, the problem is unsolved even for measure spaces. Since the condition $,$ = $42 gives us a further restriction on the map $, the conjecture might conceivably be false for measure spaces, but true for ergodic dynamical systems, for example. This also is unsettled. and O2J(9,to say that Ol and O2 It has become customary when 011Q2 are weakly isomorphic. Example Let us continue with Example 3 of Section 1. Recall that O was the one-sided shift on k points, with X = X.",, X , , and 0' was the two-sided shift, with X ' = X,"= - X , . Thus W is invertible, but Q is not. Define +: X ' -+ X by 1(1( ..., x - ' , x o , xi,. . .) = (xl, x2,...). If

c = {x E x : (X,,,

. . ., x,,)

E

A)

(24)

24

I.

ERGODIC THEORY

is an arbitrary cylinder set in X, then $-I(C) has the same description with X replaced by X’. Thus $I, is measurable and measure-preserving. Clearly 4$ = I)$’, so that @lo’.In particular, it follows from the following proposition and the discussion at the end of the preceding section that @ is strongly mixing. Proposition 1.7 Suppose that Q1 I@ and that @ is (1) ergodic, ( 2 ) weakly mixing, or ( 3 ) mixing. Then OIhas the same property. = (X, p, 4), where is Proof If we represent O1 in the form an invariant sub-a-algebra of 93,then the relation given in (13), (14), or Definition 1.5 is true for all A, B E 9,hence, in particular, for all A, B E B1. I

5. INVERSE LIMITS

The direct product of infinitely many dynamical systems may be thought of as a limit of finite products in a way which will become clear in the following. On the other hand, the slightly more general notion of inverse limit is also useful in the calculation of entropy (Chapter IV) and the analysis of complex dynamical systems. Rather than a constructive definition, which is possible for the inverse limit of a sequence, we shall give a categorical definition of inverse limit, thus avoiding temporarily some of the sticky problems of existence. That is, our definition will involve only homomorphisms between dynamical systems and the completion of certain commutative diagrams. We note in passing that the direct product could also have been defined categorically. Recall that a set J is said to be directed if there is defined on J a relation < such that (i) < is a partial order on J, and (ii) for each pair a, p E J there is a y E J such that a < y and /3 < y. Definition 1.9 By an inverse system of abstract dynamical systems we shall mean a triple ( J , O U ,$?@) such that J is a directed set; for each a E J, Ouis an abstract dynamical system; and for each pair a, p E J with a < 8, we have +us : OP-+aa.An upper bound for such a system is a dynamical system @ with a set of homomorphisms pz : @ --* OZ (a E J) such that for each a, p E J the diagram

5.

25

INVERSE LIMITS

commutes. Finally, an inverse limit 6 of the inverse system (5, CDa, t,ba,J is an upper bound with maps $a : CD + CDa which is a factor of every other upper bound. That is, whenever CD is an upper bound with maps p, : CD + mu, there exists a homomorphism o: CD + 6)such that the diagram

commutes for each a E J. In this case, we write

6 = inv lim aeJ

(6, 9,) = inv lim(ma,

or

a€J

Clearly, if CD = (X, a, p, 4) is an upper bound for the system ( J , CDa, +as), then we can represent the ma as (X,B e , p, +), where the 99, (ct E J ) form an increasing net of invariant sub-o-algebras of .c#. The mappings t,bap and p, then become the identity mapping on X. Moreover, the inverse limit 6 can be identified with (X, 3, p, 4), where & is the smallest o-algebra containing aa.This is true because 3 reappears as an invariant sub-o-algebra of a for any upper bound 0, so that 6 = (X,d, p, 4) is a factor of CD and the commutativity of (26) is trivial. In fact, this argument shows that any bounded system of abstract dynamical systems has an inverse limit, and that all such inverse limits are equivalent in the sense of Definition 1.2. Thus we have proved the following theorem.

u,

Theorem 1.7 If ( J , a,, t,bap) is an inverse system of abstract dynamical systems, and if@ is an upper bound with maps pa, then

6 = (x,d, p, 6) = inv lim ZE

J

where & is the smallest o-algebra containing all of the p i '(a,). In particular, the inverse limit, when it exists, is uniquely determined up to equivalence.

The question of existence of the inverse limit is somewhat more difficult. The usual approach is to define the inverse limit set

I

X, = x

E

X X, :

acJ

x p = x, for all a, p E J , a < fi},

(27)

define the projections pa : X , + X u in the obvious way, and attempt to p,- '(99#)to the a-algebra am extend the measures p, pa from 99,, =

ua

26

I.

ERGODIC THEORY

generated by B o . However, it is known (see, e.g., [31, p. 2141) that this is not always possible. For the most part, we shall be interested in inverse limits only when we have an explicit representation. However, the following theorem is not without interest. The proof [12, 141 is omitted.

Theorem 1.8 Let ( J , O a , be an inverse system of abstract dynamical systems. Then there i s a system ( J , Oa’, $ip) such that Oa’ is equivalent to tDa ( c i J~ ) under a set of equivalences which carry the into I& and such that = inv lim Oa’ usJ

exists. Moreover, Oa is dejined on the inverse limit set (27) for (J,@a’,

#is).

We shall see several examples of inverse limits in Chapter 111. (See also Exercise 23.) For now we consider only two simple, but important examples. The first and most obvious is the direct product defined in the preceding section. For this we let I be the set of finite subsets of J, directed by set inclusion. Then @I Oa = inv lim J ( a , , . . ., a,) E

uE

QU,

I

0 QU2 0

0 (Dan.

The maps fiu and $uB are the obvious “ finite-dimensional” projections, and a routine verification shows that the appropriate diagrams commute. If @ is any other upper bound, then the map c: O + B U E Oa, J given by c ( x ) = ( ~ ~ ( Jx, completes ) ) ~ ~ diagram (26). As an application of Theorem 1.8, we give the following construction due to Rohlin [51]. If tD is a noninvertible dynamical system, it is possible to define an invertible system &, called the natural extension of 0, such that Q, is a factor of 6,and 6 is a factor of any other invertible system of which (3 is a factor (see Proposition 1.9 below). For each positive integer n let Q,, = Q, and @,,, = i$m-n for m > n. This defines an inverse system indexed by the set J of positive integers. Let & = inv limns Q,, = inv limn+a Q,,, . Taking & = Omas in Theorem 1.8, and noting that we can write m

X,

= (x E

)( X , : x ,

n= 1

= i$(x,+

for each n},

we see that &1,

x2 x3, * ..) = ( i $ ( X , ) , 9

4 4 4 , i$(x3),. . .) = ( i $ b l ) , x1, x2

9

* *

-1.

5.

Thus

27

INVERSE LIMITS

# is one-to-one, and its inverse

6- yx1, x2, x3, * . .) = (x2

3

x3 9

x 4 , * * .)

is also measurable. That is, 61is invertible. Of course, if O is invertible, then d) is isomorphic to 0. In fact, p 1 is an isomorphism, since x,, = (6-"x1 for x E 8. Proposition 1.8 Let ( J , Oa, $uB) be an inverse system of dynamical systems, and let J , E J have the property that for each ci E J there is a p E J , such that ci i p. Then ( J , , Ou,$uo) is an inverse system, and inv lim Ou= inv lim Ou. a E J

a E Jo

Proof This follows from the corresponding property for a-algebras. Thus, if Ou= (X, Bu, p, 4), it is clear that &?a = U a E J , Bu, and the result follows from Theorem 1.7. I

UuEJ

The proofs of the following two propositions are routine verifications and will be omitted. Proposition 1.9 If

to : Za -+Oufor

each

ci

E

J , then

inv Iim(C,, map)[ inv lim(Oa, $uB)r a E

J

a E J

provided that the diagrams

commute for each a, E J . I n particular, if 210, then overcarat denotes the natural extension.

216, where

the

Suppose that Ol and a2 are factors of O. Then we may write Ok= (X,Bk,p, 4 ) ( k = 1, 2), where 0 = (X, LB, p, (6). Let us denote by Bl v B2the smallest o-algebra containing both and B2.We define the join of O1 and O2 to be v (D2 = (X,Bl v B2,p, 4). Of course, the notation and terminology extends to joins of arbitrary families of sub-a-algebras of B and of factors of O.

28

I.

ERGODIC THEORY

for each a E J , then

Proposition 1.10 If 0,' and OU2are factors of inv Iim(Q,,' v U E

J

aU2) = (inv ~ i m a,') UEJ

v (inv lim Q,,'), U E

where the latter join is as factors of inv lim,

J

0, . In particular,

inv 1im(Qu'@ a,*) = (inv Iim Q , ~ ' ) (inv Iim a,,"). acJ

UEJ

UEJ

Proposition 1.11 The inverse limit inv Iimue Q,, = Q, is (1) ergodic, ( 2 ) weakly mixing, or (3) mixing ifleach Qu has the same property. Proof Since each OUis a factor of 0, the result follows in one direction from Proposition 1.7. To prove the converse, let us denote QU = (X,W,,p, 4 ) where Q, = (X,W, p, 4). According to Theorem 1.7, the algebra B0 = U u EW,is dense in 9. Thus (see Exercise 19) condition (13) or (14) or the defining property of mixing holds on A9 iff it holds on Wo. But the latter is true iff it holds on each W,. I Corollary The natural extension of or (3) mixing i@Q, is.

0.

Q,

is (1) ergodic, ( 2 ) weakly mixing,

INDUCED SYSTEMS

In 1943, Kakutani [36] introduced the idea of a transformation induced by a measure-preserving transformation q5 on a subset A of positive measure. The idea is to localize the system and only observe @ ( x ) when it is in A. This has been a very fruitful idea for constructing examples and has recently begun to play a role in the theory of abstract dynamical systems somewhat analogous to that of factor systems. The basis of the construction is the recurrence theorem of Poincare (Theorem 1.5). Thus if 4 is a measure-preserving transformation on a finite measure space (X,8, p), and if A E W is a measurable set of positive measure, then for almost every X E A there is a positive integer n = nA(x)such that @ ( x ) E A, but q5(x), $,"(x),. . ., q!P-'(x) q! A. Definition 1.10 The induced transformation on a set A E W with p ( A ) > 0 is the transformation 4, : A + A defined by 4 , ( x ) = @"'(")(x),where n,(x) is the smallest positive integer n such that @ ( x ) E A. The induced dynamical system is a, = (A, a,,p A , 4,), where W A= {A n B : B E 93} and p A is the normalized (total measure one) restriction of p to W,.

6.

29

INDUCED SYSTEMS

Of course, 4 A is in general only defined for almost all x E A. Its definition may be extended arbitrarily to all of A.

Theorem 1.9 The induced transformation 4 Ais measure preserving. Thus O Ais an abstract dynamical system. Zf O is invertible, so i s O A. Proof

Define for n 2 1 A, = {x E A : n A ( x ) = n} = { x : x, @(x) E A ; $(x), . . ., @- 1(x) 4 A } B, = { x : X, 4(x), ..., @ - ' ( x ) $ A;&"'x)E A}.

Since 4 is measurable, we have A,, B, E W. Moreover, A, E A and for each C E a

4;1(~) =

6

n= 1

[An n +-Yc)I.

(28)

It follows that 4 A is measurable on (A, W A ) . Now the sets A, ( n = 1, 2, . . .) form a disjoint partition of (almost all of) A, and the sets B, ( n = 1, 2, . ..) form a disjoint partition of the set of all points whose "orbits" intersect A minus A (almost all of X A in the case of an ergodic 0).Also 5-

f#rl(A) = A , u B ,

For any C E that

(29) (n 2 1). 4-'(Bn) = An+1 u Bn+1 with C E A, it follows by repeated application of (29)

4- ' ( C )= [ A , n 4- '(C)l LJ [Bl n 4- Wl n 4-("+')(C)], 4-'[B, n 4-"(C)] = [ A , + , n $-("+')(C)]u or, since 4 is measure preserving, n

~(= c k)1 p(Ak n +-'(C)) + p(Bn n 4-"(C))* = 1 Since the B, are pairwise disjoint, the last term tends to zero. Thus from (28) 5

P(C)=

C1p(An n # - " ( C ) ) = ~ ( 4' (,C ) ) .

n=

Now suppose that O is invertible. Then, of course, 4-l is measure preserving, and we can define (4-l)", the induced transformation on A. We shall show that ( 4 - 1 ) A = (4J1. By symmetry it is sufficient to show

30

I.

ERGODIC THEORY

( 4 - ' ) " ( 4 " ( ~=) x) for almost all x E A. Suppose that x E A,. Then +"(x) = @"'x) = y E A.Clearly, 4-"(y) = x E A. Suppose that z = +-"(y) E A for some m, 1 Im .c n. Then 4"(z) = y = 4"(@"'"(x)). But @"'"(x) # A,

that

and this contradicts the fact that 4" is one-to-one. It follows that ( 4 - ' ) " ( y ) = $-"(y) = x, as was to be shown. I

Figure 1. (a) Induced transformation; (b) inverse construction.

Proposition 1.12 If@ is ergodic, so is 0". Proof Suppose that C E B,C E A, and

4;

'(C) G C. Define

W

D

=

U {[An n 4-"(C)l u [Bn n 4-"(C)l)*

n= 1

According to (28), A, n 4-"(C) C C for each n, and so A the other hand, by (29)

-

C E d. On

# - ' [ A , n 4-"(C)] C $ - ' ( C ) = [ A , n 4-'(C)] u [El n $ - ' ( C ) ] and

4- ' [ B , n +-"(C)]= [ A , , n q!-("+ "(C)] u [ B , + ~n +-("+')(C)]. Thus 4 - ' ( D ) c D . Since 0 is ergodic, either p ( D ) = 0 or p(d) = 0. Suppose

-

that p ( D ) = 0. It follows from (28) that p ( 4 ; ' ( C ) ) = 0, and hence that p(C) = 0. Likewise, if p(d)= 0, then from the preceding p ( A C) = p A ( e )= 0. It follows that 0"is ergodic. I

In case 0 is invertible, there is an interesting way of describing the transformation 4 in terms of 4" and the sets E n . This will lead to another new construction, which is, in a definite sense described below, the inverse of the induced transformation construction.

6.

31

INDUCED SYSTEMS

u:=,

-

Let us write B, = A, so that X = B,. Note that maps B,+l onto a subset of B, for each n, and that 4-l maps x E B, onto the point 4; ‘+“(x). Now suppose we are given a disjoint sequence of sets B, E W,where (X, W,p ) is a finite or cr-finite measure space. Suppose further that p(B,,+1) I p(B,) < 00 and p(B,) + 0 as n + 00. For each n let a, : B,, + B, be an invertible measure-preserving transformation of B,+ onto a,(B,+ l). Let 4, : B, + B, be an invertible measure-preserving transformation of B , onto itself. We define a mapping 4: Y + Y, where Y = B,, by

u:=,

Theorem 1.10 The mapping 4 is an invertible measure-preserving transformation. If p( Y ) = 1, Q, = ( Y , W,p, 4) is an abstract dynamical system, and 4, is the transformation induced b y 4 on B,. If 4, is ergodic, so is 4. We leave the proof as an exercise. Suppose Q1 and Q2 are invertible, ergodic dynamical systems. Let us induced by O2 on some is isomorphic to a system write Q1 < O2 if set A of positive measure. In Kakutani’s terminology, a1 is a derivative of Q 2 , and Q2 is a primitive of There is a clear analogy to the theory of factors discussed in Section 4, and again the question arises as to whether Ol 4 a2and a2< O1imply a1z 0 2 . The construction preceding Theorem 1.10 may be described by saying that Q, = (Y, 93,pi 4) is constructed on the system Q,, = ( B , , W,,p o , 4,). From the discussion it is clear that this is equivalent to Q,, < Q,, at least when Q, is ergodic. A discussion of this in terms of “flows under a function” is given in the exercises. Example (Kakutani [37]) Let B, be the unit interval with Lebesgue measure for p. (We can take, for example, X = R x Z to be the product of the reals with the integers.) Define Cpo on B , by mapping the left half of the dyadic interval [1/2”, 1) linearly onto the right half: +o(x)=x-

1 1 1+-+2”+” 2”

1 1- - < x < 1

2” -

1 -2”+1,

Let B1 be a linear set of length $ “sitting above”

n = 0 , 1,2,....

32

I.

ERGODIC THEORY

and let B, = @ (n > 1). It is easily seen that q5,, and therefore also #, are ergodic. A little more effort (Exercise 35) reveals that (Do has discrete spectrum, that is, T+ohas enough eigenfunctions to span LJB,). According to Theorem 1.6, (Do is not weakly mixing. On the other hand, (D is weakly mixing, but not strongly mixing (Exercise 36). Thus Proposition 1.12 fails if “ergodic” is replaced by “weakly mixing.”

Figure 2. Kakutani’s example.

EXERCISES Measure-Preserving Transformations

1. (a) If V is a class of subsets of some set, let a(%?) denote the denote the smallest smallest algebra of sets containing W, and let a(%) a-algebra containing W. Suppose that ( X i , W i , p,) ( i = 1, 2) are finite measure spaces, and that 4: X, X,. If V E a2with 93(V)= W 2 ,and if #- ‘ ( B )E Wlfor all B E V, show that 0 is measurable. (b) If, in addition, V satisfies A, B E 59

A

-

B is a finite union of pairwise disjoint sets in 59.

(30)

and if p1(q5-l(C))= p2(C)for all C E W, then q5 is measure preserving. (c) The class V of measurable rectangles in a product space satisfies (30). 2. If (X, 93, p) is a a-finite measure space, we define measure-preserving transformations of X in exactly the same way as for a finite measure space. Does #(x) = x + 2 define a measure-preserving transformation of (i) the reals with Lebesgue measure, of (ii) the positive reals, of (iii) the integers withcountingmeasure?Howabout d(x) = 2x?Show that q5(x, y ) = (2x, y / 2 ) is a measure-preserving transformation of the Euclidean plane.

33

EXERCISES

3. (Baker’s transformation) Define 4 on the unit square [0, l] x [0, 11 by 4(x, y) = (2x, y/2) for 0 I x < and 4(x, y) = (2x - 1, (y + 1)/2) for

4

4 I X I l . (a)

Show that

4 is measure preserving.

(b) By mapping the sequence {x,} of 0’s and 1’s onto the point

... and y the expansion 4 is equivalent to the two-sided shift on two

(x, y) such that x has the binary expansion .xoxlxz . X - ~ X - ~ X - ~, show

that

points. 4. Verify Example 2 and show that it is equivalent to the one-sided shift. 5. (Adding machine transformation) Define m

( X , 9, p) =

x

( X , 9, P,)? 9

n= 1

7

where X, = (0, 1, ..., k,}, 9, is the class of all subsets of X,, and Define 4 : -+ by pn = { P n o , Pni, +

x x

’ 9

4(x1, x 2 , ...) = (xl =

+ 1, x z , x 3 , ...)

(0, ..., 0, x,

+ 1,

X,+l,

if x1 < k , xp+2, ...)

if x1 = k,, .. ., xp- = k,-,,

x, < k,

4 ( k l , k 2 , . ..) = (O,O, 0, ...).

Show, as in Example 3, that the inverse image of a cylinder set is a cylinder set. Conclude that 4 is measurable, and that it is measure 1). preserving iff p n j is independent of j ; namely, pnj = l/(k,

+

Doubly Stochastic Operators 6. If (X, 9, p) is the unit interval or one of a certain class of “decent” measure spaces, then, for each set function +: &?+.9if which preserves finite and countable unions and intersections and also preserves complements, there exists a measurable point transformation 4 : X -+ X such that + ( B ) = #-‘(B) for all B E %. Thus Proposition 1.2 may be proved by exhibiting such a .)I (a) If ,yA is the characteristic function of the set A, show that for T a doubly stochastic isometry on L2 and for any A, B E %

r‘I

(TXA)(TXA

dp =

[

‘ X

TXA n

Bdp.

(b) Show that 0 IT x AI 1 and hence that 0 I (Tx,,)’

ITxA.

34

I.

ERGODIC THEORY

(c) Use (a) and (b) to show that (TxA)’= TxA, and hence that TxA is the characteristic function of some set $(A). (a) Show that T z A I min{TX,, TxB},and hence that

0I TxA

= (TxA n

d2

(TxANTxB). (e) Use (a) and (d) and the relations x A U B = x A ze - x A n B , zn - 1 - x A , to conclude that $ preserves finite intersections, finite unions, and complements. ( f ) From p ( A ) = ( x A , 1) deduce that I) preserves measure and hence also countable unions and intersections. 7. Suppose that T is an operator on L , ( X , 33,p) where p is a finite or a-finite measure, and suppose T satisfies (i) f S O * Tf 2 0, (4 IITf Ill 5 (3 IIT f I1,5 I1f 1I m where f E L, for (i) and (ii), and f E L1 n L, for (iii). Suppose further that gE L, n L , . (a) Show that (Tg - c)’ 5 T(g - c)’ for any constant c. (b) Show Sx (Tg - c ) h(Tg, c ) d p s jx(g - c ) h(g, c ) dp, where nB

+

Ilflll7

9

h(u, u ) = 1 =O

if u > u ifulu.

(c) Suppose that g 2 0. Multiply both sides of the above inequality by cP-’ and integrate with respect to c from 0 to co. Apply the FubiniTonelli theorem to obtain

and hence IlTgIlp 5 l l g l l p . (a) From (c) and ITg(I Tlgl deduce that llTllpI 1. 8. (a) Let T be a doubly stochastic operator. Suppose that 0 If, 7 f a.e. with f E L,. Show that Tfn7 Tf by showing that S B Tf d p = j B limn+, T L d p for each B E B. (b) The preceding is a “monotone convergence theorem” for T. Formulate and prove a “dominated convergence theorem” for T. Ergodic Theorems 9. Let (X, g,p) be a a-finite measure space. The statement and proof of Theorem 1.1 remain valid in this context. (a) Show that Corollary 1.1.1 also remains valid as follows. Let

EXERCISES

35

fn*(x)= maxlsksfi(l/k)&f ( x ) and A,*(f; a ) = {x : fn*(x)> a}. For fixed n and any measurable set C with finite measure, let h = f - axc. Deduce as in the proof of Theorem 1.1 that f&*(h)h d p 2 0 so that

(b) If {C,} is an increasing sequence of measurable sets with finite measure and union X, show that as j cc the sequence B,*(f - axe,) decreases to A,*(f; a ) if a > 0 or increases to the same limit if a < 0. Conclude that --+

and complete the proof of Corollary 1.1.1 by letting n -+ 00. (c) The proof of Theorem 1.2 now goes through as before. In particular, j 1 f 1 d p s j 1 f I dp. Show that equality does not always hold by considering the transformation 4(x) = x + 1 on the reals. 10. (a) By an appropriate choice of the function g, show that Birkhoffs theorem follows from the Chacon-Ornstein theorem (Theorem 1.3). (b) Let P = ( p i j ) be an infinite matrix with E j p i , =. 1 for each i and p i j 2 0 for all i, j . Let Z be the integers, and provide it with a measure structure by letting p ( A ) be the number of elements in A. Define T by T(f,} = {gi}, where g j = X i p i j f j . Show that T satisfies the hypotheses of Theorem 1.3. Conclude (Kolmogorov’s theorem) that

exists, where pi;) is the (i, j)-entry in Pk.Also (ratio limit theorem)

exists. 11. (Mean ergodic theorem) Let @ be an abstract dynamical system and T = T6. (a) Suppose T*F = F.Byevaluating IITF - F1Iz2= (TF - F, TF - F), show directly that TF = F. This gives a simplified proof of Theorem 1.4 for p = 2. (b) I f f E L,, then Theorem 1.4 implies that (l/n)T,f -,f * in measure. O n the other hand, by Theorem 1.2, ( I / n ) T , f - + f in measure. Hence f = f * a.e. In particular, f~ L,.

36

I.

ERGODIC THEORY

(c) Show from (b) that Jx f d p = Jx f d p . This can also be proved directly by considering the restriction of 4 to the invariant set

B(a, p) = {x : a c f(x) IP) and applying Corollary 1.1.1 to obtain @P(B(%PI)

W ( N %P))

1 j *

B(a, 8 )

4~~ 8)

f

dP 5 PP(B(% P ) )

f dP 5 BP(B(a9 B)).

In particular,

Adding on k = 0, 1, i 2 , . , . and then letting n --* co gives the desired result. 12. Show that f * is an invariant function (in Theorem 1.4); that is, show that Tf* = f*. Recurrence

13. (a) In Theorem 1.5 show that almost every point of A returns to A infinitely often. (b) Show that the conclusion of Theorem 1.5 fails for the transformations defined in Exercise 2. 14. (a) An operator T on L, is said to be conservative if f E L , , f 2 0, C:=o T"f(x)< cc a.e. implies that f = 0. Show that any doubly stochastic operator on L, of a finite measure space is conservative. (b) If 4 : X + X is any measurable transformation on a a-finite measure space, and if T&is conservative, then the sequence A, 4-'(.4), @-'(A), . . . can be pairwise disjoint only if p ( A ) = 0. Hence 4 fulfills the conclusion of Theorem 1.5. Ergodicity and Mixing 15. If (X, A9, p) is a finite measure space, and if 4 is a measure-preserving transformation, show that 4- '(B) E B implies 4- ' ( B ) = E . Thus 4 is ergodic iff 4- '(E) E E G- p ( E ) = 0 or p(B) = 0. Show that the two definitions

EXERCISES

37

are not equivalent in the case of a a-finite measure space. We adopt the latter as our definition of ergodicity in that case. 16. Let Q, be an abstract dynamical system. Show that the following are equivalent : (a) Q, is ergodic. f is a constant. (b) q f = f , f E L, (c) For all f E L,, 1 n-1 a.e. lim - C T,kf(x) = f dp

(a)

For all f

E

[

' X

nk=O

n-m

L,, X

(e)

For all f

E

L,, g E L,, where l/p + l/q = 1, lim

n+m

;1 ( f , q k g ) k=O

= ( f , 1)(1,g).

(f) For all A, B E 2,

1n-1

lim

n-+x

(g)

n c Pc(A n 4 - k ( B ) )= P(A) P(B). k=O

For all A, B E 2 with p ( A ) p ( B ) > 0,

1 4 ' 4 n 4 - " ( m '0. X

n= 1

(h)

For all A, B E J with p ( A ) p ( B ) > 0,

1p ( A n +-"(B))= + 00. X

n= 1

17. Let Q, be an abstract dynamical system. Show that the following are equivalent : (a) Q, is weakly mixing. (b) For all f , g E L , there exists a set J of density zero such that

lim

n-x.n$J

(c)

For all f , g E L,

( f , Tbng)= ( f , I)( 1,g).

38

I.

ERGODIC THEORY

18. Let (bo be an ergodic measure-preserving transformation on [0, 11. Let X consist of the two disjoint line segments X, = {(x, 0) : 0 5 x I 1) and X 2 = {(x, 1) : 0 I xI 1) with linear measure normalized to one. Define (b on X = X1 u X, by (b(x, 0) = (x, 1) and (b(x, 1) = ((bo(x), 0). (a) Show that (b is an ergodic measure-preserving transformation. (b) Show that p(X1 n 4-"(X2))takes on only the values 0 and hence does not converge. 19. (a) If d is an algebra of subsets of X, and if .4? = B ( d ) (see Exercise l), then for each A, B E &? and each E > 0 there exist sets A , , Bo E d such that

4,

I@

n 4-k(J4) - P(A0 n (b-k(Bo))I 5 P[(A n 4 - k ( B ) )A

(A0

n 4-k(B0))1

Ip [ ( A A A,) u ((bPk(BA Bo))] < 2~ for all k . (b) If %j is a class of subsets of X satisfying condition (30) of Exercise 1, and if .4? = W(%j),then Q, is (1) ergodic, (2) weakly mixing, or (3) mixing iff the defining relation is satisfied for all A, B E W.

Products and Factors 20. Give an example of ergodic systems (Dl and 0,such that 0,0 a2 is not ergodic. 21. Show that the union of a finite number of sets of density zero has density zero. Show that this is false for a countable number. 22. If W ,c W,show that 0 = (X, p, 4) is an invertible dynamical system iff (b-'(&?,) = gl. Inverse Limits

23. Show that the system (D of Exercise 5 is an inverse limit of the sequence 0,= ( K , d n v,, , 4,), where

X' and

n

n

xxk,

d n =

k= 1

4n(x1, *.., xn)

+ 1, ~

=(XI

x k=

n

v,=

.4?k? 1

)(&,

k= 1

2 . ., - ,xn)

if x1 < k , = (0, . .., 0, xg

+ 1,

Xp+,,

*

.., x,)

if x1 = k , , ..., x p - = k p - , , (b"(k1, k 2 * . ., kn) 9

=

(O,O,. . . >0).

x p c k,

39

EXERCISES

24. A Lebesgue system is an abstract dynamical system @ = ( X , a, p, 4) such that there exists a countable class %? G 93 with 93(%') = 93. Show that the inverse limit of a countable number of Lebesgue systems, and hence also the direct product of a countable number of Lebesgue systems, is a Lebesgue system. 25. A Kolmogorov system is an invertible dynamical system @ = (X, 93,p, 4) for which there is a o-algebra goG satisfying (i) 4-'(@,,) c Bo, 4-n(9%,)= X } , and (ii) = 93. (iii) g(lJ,"=@ao) An exact dynamical system is a dynamical system @ = ( X , 93, p, 4 ) satisfying (iv) 4-"(9) = Show that @ is a Kolmogorov system iff it is the natural extension of an exact system. 26. Show that the two-sided shift on k points is isomorphic to the natural extension of the one-sided shift on k points. Show also that the two-sided shift is Kolmogorov. 27. Prove Propositions 1.9 and 1.10.

{a,

{a,x}.

Induced Systems 28. Show that Theorem 1.9 and Proposition 1.12 remain valid if 4 is a recurrent (i.e., one for which the recurrence theorem is valid) measurepreserving transformation of a a-finite measure space. 29. Prove Theorem 1.10. 30. Construct an example of an ergodic measure-preserving transformation on the reals. Show that any such transformation is conservative, hence recurrent. 31. If is induced by @ = ( X , 93, p, 4) on A, then @ A is also 4 - " ( A ) is a &invariant induced by 0'= ( Y , By,p y , $ y ) , where Y = subset of X . Moreover, Y is the minimal subset of X for which this is true. (a) Show that @ A is ergodic iff (P' is ergodic. (b) Show by example that (a) is false if "ergodic" is replaced by "mixing" or "weak mixing."

u."=o

Special Flows

32. Let @ = (X, a, p, 4) be a dynamical system, and let f 2 0 be a nonnegative measurable function defined on X . Let Y be the space under the graph of f , that is, Y = {(x, y ) : x E X , y E R, 0 Iy < f ( x ) i . Y inherits

40

I.

ERGODIC THEORY

a measure structure as a measurable subset of the product space X x R. Define a family +,, 0 I t < 00, of transformations of Y as follows: 4,(x, Y ) = (x, Y

+ r),

0 I JJ

+ t < f(x)

k=D

c f(4“4)I Y + < c f(4k‘x’).

n- 1

k=O

n

t

\

k=O

If t is thought of as time, the point (x, y ) moves upward with velocity one until it reaches the “roof” of the space Y , then moves back to the “floor” X and is transformed by 4. (a) Show that (b, is a measure-preserving transformation of Y for each t 2 0. (b) Show that the transformations 4, form a Jow in the sense that (b, 4s= d,,, for each t, s 2 0. (c) Show that the flow is measurable, in the sense that 4.(.): Y x R + --* Y is a measurable function. The flow defined in Exercise 31 is called the special Jow constructed under the function f on the system 0.Special flows were introduced by Ambrose [5], who showed that all ergodic measurable flows are isomorphic to special flows. 33. In the construction preceding Theorem 1.10, letfbe the step function defined on B, by setting f ( x ) = n + 1 if x is in the range of boo1 on-l,but is not in the range of oool on (x lies “under” B, but not and f ( x ) = 1 if x is not in the range of o o . If JI, is the special flow constructed under f on O 0 , show that the 4 of Theorem 1.10 is isomorphic to a factor of i,hl. (In the construction of the flow JI,, look at the subalgebra of sets generated by vertical “columns” between floors.) Does this imply that every ergodic transformation can be embedded in a flow? 34. If a1
35. (a) Let (Do be as in Kakutani’s example at the end of Section 6. Show that O0 is isomorphic to the adding machine of Exercise 5 with k, = I for each n. Deduce from Exercise 23 and Proposition 1.11 that Q0 is ergodic but not weakly mixing.

41

EXERCISES

(b) Set fo E 1, and, for each k = 1, 2, 3, . . ., 2"-' and n = 1, 2, 3, ..., define fk, ,on the space of Exercise 5 by fk. "(xi.x2, . . .) = exp(2ni(2k - 1)2-'"+ ')

n

C 2jxjJ.

j= 1

Show that fk,,is an eigenfunction of T+owith eigenvalue

Note that the range off,,, is just exactly the 2"th roots of unity in some order, and that 2,'- ,is also a 2"th root of unity. (c) Show that fo and the fk,,'s constitute a complete orthonormal system in complex L2(B0). [Hint: Show first that this is true for each L,(X,) as defined in Exercise 23.1 (d) For an arbitraryf E L,(Bo),expandfin a Fourier series with respect to the basis of (c). Use this expansion, Parseval's relation, and the identity A;,",,= 1 to deduce lim n-m

S,'f(42"(x))- f(x)12 ,u(dx)

= 0.

36. Let 0 be as in Kakutani's example. Note that in the terms of Exercise 35 the set A over which B1 is constructed is the set of x = (x,,} for which the smallest n with x, = 0 is odd. (See Fig. 2.) Define u, on Bo by u l ( x ) = z A ( x ) + 1, and n- 1

un(x) =

1

k=O

~1

(4Ok(x)),

x E Bo.

Note that un(x) is n plus the number of "visits" to A in the orbit x, 40(x), ..., &-'(x). For n = 4,, regardless of what x is, the first 2p coordinates y,, . . ., y,, take each of the 4, possible combinations of values exactly once as y = 4 o k (traverses ~) this orbit, y,, increases by 1, and no other change occurs in I;, j > 2p. For all but one of these combinations, namely 1, 1, . . ., 1, y E A or y 4 A regardless of the values of yj, j > 2p. In the exceptional case, y E A for exactly 4 of the x values. (a) Show that u,(x) for n = 4p takes only two values, a, and ap + 1, where

and that

p0(x : u q r ( x )= a,} = 4.

po{x : U ~ ~ ( X=) a,

+ 1) = 3.

42

I.

ERGODIC THEORY

(b) Show that 40"(x) = 4"n'"'(x)

for each n = 1, 2, . . . and each x E B o . (c) Iff E L , ( X ) is such that f(+(x)) = e2.'Y(x) for all x E X,show that

Conclude from Exercise 35(d) that Iz = 0. According to the converse of Theorem 1.6(ii), 0 is weakly mixing. (d) Use (a) and (b) to show that B~ n ~ - o P ( C E )4o4'(c)u 404'(4(C)) for any measurable set C. In particular, if C = {x E Bo : x1 = 0}, then the right-hand side reduces to 404'(C) = c.

Therefore, p((Bo

for all p = 1, 2, ..., and

-

C ) n d-"p(C))= 0

4 is not strongly mixing.