# An introduction to ergodic theory - Walters P.

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m(A)m{B) inf I —-I < m{A n B) < m(A)m(B) sup I —-

‘•j \ Pj J fj \ Pj

a £ N a £ iV

The collection Jl is a monotone class and contains the algebra

au Uj a + r

U UV T~nsf{£)

и — Л’ г — О и — a

and hence contains the c-algebra N/«x=.v T~ns/(<;) = T~NJC Let e > 0 be given and choose t0 so that t > r0 implies

112 ■! tniro'py

for .iii i. /. Then if li £ V!: J I " A I wc lr.ivc ni{A о li) —

\vhcne\ cr .1 e I

■ V

-1)

'.У and \ is 1

л г sic. In paruculai this

N

is iruc for

1 v- ;_n T ' /, Nov, ii\ л e P.v и I and consider the collection 4

of all measurable sets В with jm{A n В) — /н(.-1)/н(В)| < The collection .Л is a monotone ciaoa and contains \/*-£ T~"-r-/(c,) for all b and all .s > 0. Therefore Jf = .rt so that \m(A n B) — ni(A)m(B)\ < r. for all В e /А and all .1 e , !„'=0 '. We then gel т\Л n B) = m(A)m(B) for ali fie J and

all .-1 £ fV=o T~nJT, so by putting В = .4 wc have m(A) = 0 or 1 whenever

□

(3) One can generalize the notion of a finite-dimensional torus to obtain another kind of homogeneous space called a nilmanifold. Let ;Y be a connected. simple connected, nilpotCnt Lie group and D a discrete subgroup of ,Y so that the quotient space N/D is compact. \ D is called a nilmanifold. When V = R" and D — Z" wc get an n-iorus. The Haar measure on Лт determines a normali/.ed Borel measure on SjD. 1 f A: .V —> N is a (continuous) automorphism with AD = D then ihi> induces a map 1: N/D —- N/D, which i.s called in uuiiHiwrphism of N’D. The automorphism A always preserves the measure in. Parry has investigated the ergodic theory of such mapt and has shown that if A is ergodic then A is a A'-automorphism. A subclas. of the ergodic automorphisms of N/D arc known lo be Bernoulli automorphism^ (see Marcuard [1]).

The simplest examples are as follows: Lei

h x z\ 1

= - 0 1 ,•

\0 0 1 j J

.V satisfies the above conditions with ihe operation of matrix multiplication and the natural topology from Ri. Let ,

/1 in p\

D = J 0 1 n : m, iiy p e Z\

1 \0 0 1j

Then N.D i.-> a nilmanifold The automorphism

/1 i о \o

2\ + у : -r x2 + у

л h у 1

of Л' indotts an ergodic automorphism of N/D.

(4) The following is an example of a Kolmogorov automorphism that is not a Bernoulli automorphism The proof of this .s due to S. Kalikow [1].

Let T: X -* X denote the two-sided (i-, -^-shift. We shall define a transformation of the direct product measure spacc X x X. If х = (xn)C a у = (y„)I

J4.10 The Pinsker cr-Algebra of a Measure-Preserving Tunsformalion

113

where x„, \ e {0,1}, put S(x, y) = (7'v, TcMy) where i:(x) = — 1 if x0 = 0 and t:( \) = 1 if x0 = 1. Then i' is a Kolmogorov automorphism but not a Bernoulli automorphism.

§4.10 The Pinsker er-Algebra of a Measure-Preserving Transformation

Let T be a measure-preserving transformation of a Lebesgue space. Let

■S’fT) - \/{.s/:.<y с sJ finite, h(T,s/) = 0}.

This is called the Pinsker a-algebra of T.

One can show th.il T~ \'AT) = ^F(T). One can also prove that if s/ is finite then A a jF(T) iff h{T,.&) = 0. Thus /(T) is the maximum c-algebra such that T restricted to (A', ^(T),m\^{n) has zero entropy. Note that J’(T) = Л iff/i(T) = 0 and 'F(T) = Jf iff T is a Kolmogorov automorphism (by Theorem 4.34). See Rohlin [1] or Parry [2] for a full account of these results.

Theorem 4.36 (Rohlin). If T is an invertible measure-preserving transformation of a Lebesgue space with h(T) > 0 then UT has countable Lebesgue spectrum in the orthogonal complement of L2(J>{T)) in L2(26).

This reduces the study of the spectrum of invertible measure-preserving transformations to those with zero entropy.

The types of spectrum that occur for zero entropy transformations are unknown. There are examples of zero entropy transformations with countable Lebesgue spectrum (from Gaussian processes and horocycle flows). Another important result is

Theorem 4.37. Let T: X -* X be an invertible measure-preserving transformation of a Lebesgue space (X,JJ,m). Suppose FF is a sub-c-algebra of St with TJF -- JF and such that T has completely positive entropy on (A, W, m) (i.e., if s/ is finite .9/ ¥ JF and .v/ c then h(T, sZ) > 0). Then FF and 3P(T) are independent i.e. if F e ,'F and A e J’(T) then m(F n A) = m(F)m(A).

For a proof see Parry ([2], Chapter 6).

Bccause of thii theorem Pinsker conjectured that any ergodic measure-preserving transformation could be written as a direct product of one with /его entropy and one with completely positive entropy. However, Theorem 4.32(iij shows this conjecture is false because if T:A' -* A is the example of Ornstein u'ith no square root then the transformation 5 of the direct product measure space {0,1} x A (where the measure on {0,1} gives measure ^ to

4 Entropy

each poim) defined by S(0 .v) — (1,-v), S(l,.v) = (0,Tx) provides a counterexample to the Pinsker conjecture. (It is not difficult to show that consists of the four sets ф, [0} x X, {1} x X, {0,1} x X, and then one shows that if there is a sub-c-algebra S7 with SfS = rS and '£ being an independent complement for SP{S) then T must have a square root.) This example is not strong-mixing (since S2 is not ergodic) but Ornstein has also constructed a strong mixing transformation lhat violates Pinskcr’s conjecture.

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