# An introduction to ergodic theory - Walters P.

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Wrf*. uTf) = to. У-А/Тп)) = о

so gUTf„eV. □

Corollary 4.33.1 A Kolmogorov automorphism is strong-mixing.

Proof. By Theorem 2.12. □

Kolmogorov automorphisms are connected to entropy tncory by the following result (half of which was proved by Pinsker).

Theorem 4.3*4 (Rohlin and Sinai, ^ec Rohlin [3]) Let (X,n4,m) be a Lebesgue space unci let T: X —> X be cm invertible measure-preserving transformation. Then T is a Kolmogorov automorphism iff h(T, sJ) > 0 for all finite sJ ф Ж.

4 pul.-ору

Remarks

\1) One says that T has completely positive -iruropy when the latter condition holds. Hence T has completely positive entropy ilT it is a Kolmogorov automorphism.

(2; This shows that К automorphisms are "the opposites" of transformations with /его entropy (since h[T,.rV) = 0 V.c/ in the /.его cntrop) case).

(3) We a’ready know from C’orollary 4.14.4 that a Kolmogorov automorphism has positive entropy (since К r T.V).

(4) It follows from Theorem 4.34 that if T-is a Kolmogorov automorphism ilien so is T'\ We know from Theorem -r.32(ii! That T~ 1 need not be a conjugate lo T.

(5) The result-, of Rohlin and Sinai (Roll.in [3], page 37) show thaT T is a Kolmogorov automorphism iff w icnevcr ./ is J r.niLC sub-algebra of £ then Л.'=о V/=« T~J rJ = J '■ ^llc сап ьсс ^rom inis how much stronger the Kolmogorov properly is than strong-mixing because Sucheston has shown that T is strong-mixing iff for every subsequence Г of positive integers and every finite sub-alcebra s.J of /А there exists a subsequence {bj) of Г with

. \/LHT-b'.'V = .r

\6) There is another result lhai shows the Kolmogorov property is a uniform strong-mixing condition. Let T be an invertible measure-preserving transformation of a Lebesgue .-.ruce (Х,Л,т). For В e j# and к > 0 let /A(B, k) denote the smallest ст-algebra containing all the sets T~"B for n > k. Then T is a Kolmogorov automorphism iff for all A, В e Ш

lim sup |ni(4 n C) — m(A)m{C)\ — 0.

A x> Ce tf(tt.k)

In the following examples the word “automorphism” is used in two senses. When we say “group automorphism of a compact group” we mean automorphism in the sense of topological groups^ whereas in the terminology "Kolmogorov automorphism” the word autotrforphism refers to an invertible Structure preserving map of a probal^jity space.

\amplcs

(1) Group Automorphisms. Rohlin proved that any crgodic automorphism of a compact abelian metric group is a Kolmogorov automorphism [4] and later Yusinskii proved the theorem in the non-abelian case. [2]. Kat/nelson [" i j has shown that crgodic automorphisms of fmue-dimensional tori are Bernoulli automorphisms. Lind [1] and Miles and Thomas [1] have proved, using different methods that any ergodic automorphism of a compact metric group is a Bernoulli automorphism.

(2) Markov Shifts. Let T be the two-sided (p, P) Markov-shift. We have seen that T is ergodic iff P is irreducible (i.e., V pairs of states i,j 3/i > 0 with pjf > 0) and T is strong mixing iff P is irreducible and aperiodic ^.e., ЗЛ' > 0 wilh p\j1 > 0 for all states /, /'). Friedman and Ornstein [l] h ive shown that

$4.9 Bernoulli Automorphisms .md Kolmogorov Automorphisms

111

such a Markov shift is a Bernoulli automorphism. Therefore from the point of view of ergodic theory, mixing Markov chains are the same as Bernoulli automorphisms. One can easily dcduce that an ergodic Markov shift is the direct product of a Bernoulli automorphism and a rotation on a finite group.

The proof of Friedman and Ornstein consists of showing lhat a transformation with a certain property is isomorphic to a Bernoulli shift. This is a generalisation of the deep result of Ornitein (Theorem 4.28). It is however easy to show that the (p. P) Markov shift is a Kolmogorov automorphism iff P is aperiodic and irreducible. This generalises Theorem 4.30.

Theorem 4.35. The two-sided (p, P) Markov shift is a Kolmogorov uuto-morphism iff P is irreducible and aperiodic.

Proof. Let T denote the two sided (p, P) Markov shift and let (A, S), m) be the space on which it acts. If T is a Kolmogorov automorphism then T is strong-mixing (Corollary 4.33.1) and therefore P is irreducible and aperiodic (Theorem 1.27). Now suppose P is irreducible and aperiodic. Let £ = {A0,Al,... be the natural partition into states at time zero i.e.

= {{*„}-со £ A|x0 = i). Let Ж be the smallest c-algebra containing all sets in the partitions T~n£, n > 0. i.e. Ж — \J*= 0 T~"s/(z)- Then Ж <=. TJC and V» = o ТпЖ = ^ so it remains to show P|^=o Т~'Ж = {ф, A'}’. We shall do this by generalising the argument used in the proof of Theorem 4.30 Recall from Theorem 1.27 that since P is irreducible and aperiodic we have lim,,^^ p-"' = pj for all states i, j. Suppose A is a cylinder block J[i0,..., irJu + r ar>d В is a cylinder block b[y0,... with b -t- s < Sf. Then

m(A n B) = Pjlpjuji ’ P.,-.-. so thal

The same inequality is true if A e \/anZra T~"s/{^) and В e \/n=t T~"j#(38). Fix В e \Jbn=% T~nsJ(c,) and fix N > b + s. Consider the collection Jl of all measurable sets A with

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