# An introduction to ergodic theory - Walters P.

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(3) If T is a Bernoulli shift so is T2. The state .space for Г2 is (У x У,

& > .?. p x p).

(4) If 7 , and T2 arc Bernoulli shifts so is T, x T2. The state space for . T, x T2 i.-> the direct product of the stale spaces for T, and T2.

Remark. The method used to calculate the emropy of examples 6 and 8 of §4.7 i_an be used to show that if T is a Bernoulli shift then h(T) < со iff there cxiVis a countable partition ц of(Y,.?7,^) such that H(>j) < cc and -s/{ri) = -7. In ihis ca^e h(T) = H( W).

In 1958 Kolmogorov asked if entropy is a complete isomorphism invariant on iht collection of all Bernoulli shifts. This was: answered in 1969 by Ornstein (see Ornstein [1]).

Theorem 4.28 (Ornstein). Let Tg, T2 be Bernoulli shifts whose state spaces are Lebesgue spaces. If h(Tx) = h{T2) then T, is conjugate to T2. and hence isomurphic by the assumption on the state spaces (a countable direct product of Lebesgue spaces is a Lebesgue space).

The proof of this deep theorem is presented in Ornstein [1], Shields [1], and Moser et al [1]. 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 shifts to their state spaces, since the entropy depends only on the state space. It is possible, for example, for a Bernoulli shift with a state space of two points to be conjugate to a Bernoulli shift with a countably infinite state space.

Remark. Since the map (/?,, -> -У/^logp,, defined on !(/>,,...,

p,J| pj > 0 pj = 1}, has image (0, log л], for each x > 0 there is a Bernoulli shift with entropy x.

Since we are interested in measure-preserving transformations up to conjugacy we make the following definition.

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4 Entropy

Definition 4.12. An invertible measure-preserving transformation of a Lebesgue space (X,Jti,m) is called a Bernoulli automorphism if it is conjugate to a Bernoulli shift.

(The word “automorphism” is used because an invertible measure-preserving transformation is a bijective structure preserving map of a measure space)

The above remarks apply to Bernoulli automorphisms and Ornstein’s theorem says that two Bernoulli automorphisms with the same entropy are conjugate. This implies the following

Corollary 4.28.1

(i) Every Bernoulli automorphism has an n-th root. (S is an n-th root of T if S" = T.)

(ii) Every Bernoulli automorphism is conjugate to a direct product of two Bernoulli automorphisms.

(iii) Every Bernoulli automorphism T is conjugate to its inverse.

Proof

(i) Let T be a Bernoulli automorphism and n > 0. Let S be a Bernoulli automorphism with h{S) = (1 /n)h(T). Then S" is a Bernoulli automorphism with entropy k(T), and therefore Sn and T are conjugate.

(ii) Let T be a Bernoulli automorphism. Let S be Bernoulli with h(S) = !, • h(T). Then h(S x S) = h(T) and, since S x S is Bernoulli, S x S is conjugate to T.

(iii) T, T~1 are Bernoulli automorphisms with the same entropy. □

The following theorem is a summary of results about the structure of the , spate of Bernoulli automorphisms, and shows this space is closed under some natural operations on measure-preserving transformations. The proofs are given in Ornstein [1]. All the results are due to Ornstein.

Theorem 4.29

U) Every root of a Bernoulli automorphism is a Bernoulli automorphism.

(ii) Every factor of a Bernoulli automorphism isJ Bernoulli automorphism.

(iii) If {&„}* is a sequence of sub-a-akjebras of Л with T.F„ = J*,,, .7 ^ cr с • ■ •, \j 'n-x &n = jtf, and if tit? factor transformation {see §2.3) associated with each :Fn is a Bernoulli automorphism then T is a Bernoulli mm morphism (i.e. an inverse limit of Bernoulli automorphisms is a Bernoulli automorphism).

Ornstein has given a criterion fora measure-preserving transformation to be a Bernoulli automorphism. This criterion is used in tortt of the proofs of the results in Theorem 4.29 and is also useful when clitcfcing if concrete

Bernoulli Automorphisms and Kolmogorov Automorphisms

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examples are Bernoulli automorphisms. At the end of this section wc list some examples of Bernoulli automorphisms and Ornstein’s criterion (or some \ariant ol it) is the usual method used to show these examples are Bernoulli automorphisms.

Since a Bernoulli shift is really an independent identically distributed stochastic process indexed by the integers we can think of a Bernoulli automorphism as an abstraction of such a stochastic process. In 1958 Kolmogorov introduced the following class of measure-preserving transformations as abstractions of regular, identically distributed stochastic processes.

Definition 4.13. An invertible measure-preserving transformation T of a probability space (X, j0, m) is a Kolmogorov automorphism (K-automorphism) if there exists a sub-cr-algebra Ж of .JO such that:

(l) Ж <= ТЖ.

In) Vfco Т"ЛГ = JO.

f\?=o Т-Ж = Ж = {Х,ф}.

We always assume jV ^ (since if not the identity is the only measure-algeora automorphism). Hence Ж ^ ТЖ. In fact the space (X,26,m) is usually taken to be a Lebesgue space

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