# An introduction to ergodic theory - Walters P.

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There arc several other ways of stating the ergodicity condition and we present some of them in the пем two theorems.

Theorem i .5. If 7 : A' —< X is и measure-prewrvincj transformation of the probability \;>ace (X,yJ,nt) then the Jollowiiuj siattmeiU'S arc equivalent:

(il T is eri/odic.

(11) The only members В of Л with m[T~ 'В Д В) = 0 arc those with iii(/$I = 0 or in(li\ — I.

(lii) Tor i rer\ A e Jj with m[A) > 0 m haw \ T~"A) = 1.

(iv) Tor et cry /1, В e /А with m{ 1) > 0, m(B) > 0 there exists n > 0 with iiH I ! B) ^ 0.

Pkooi

(i) => (ii) Let Be.i/1 and m(T~ lB Д B) = 0. We shall construct a set Вx with T ' В, = В , and m(B Дй,] = 0. For each n > Owe have ш(7 "В Д В) = 0 bccause Т'В Д В с 7" '‘’"ВДГ'В = (Jl'-o Г'(Г‘ВДЙ) and

licncc m(T "В Д В) < пт(Т 'В Д В). Let Ви = f|'= „ К) Т~‘В. By the above v\с know m(B Д T~'B) < m(B Д T~'B) = 0 for each n > 0. Since tlic sets (_ J T ‘ В decrease with n and cach has measure equal to В we hл\ап(В , Д В) — 0 and hencc nnB , ) = m(B). Also T~ 1В , =

« , J.»„ Г .....T-''B = BV. Therefore we have

obtained ,i set В with T~ 1В —В and ш(В^ Д В) = 0. By ergodicity w-e must have пцВ,) = 0 or 1 and hence пцВ) 0 or 1.

(iij -> (ia). Let A с ,’i and нз(Л)>0. Let Ax = i T "A. We have T~ 1A i ci .3 , and since m(T ‘ A ;) = m(A J vvc have Z\Alj = 0.

By kii) we gel m(At) = 0 or 1. We fcannot have m(A ,) = 0 bccause T~>AczAl and ml 7" 1 ■!) — m(/l) > 0. Therefore m[A ) = 1.

(ni) ro (ivj. Let m{A) > 0 and п'лВ) > 0. By (iii) we have i>!('^jn'= , T~"A) = 1 so that 0 < i)i(fi) - ni(B n Ul'_ ! T~"A) = В n T~"a). Therefore

in(B r\ T~"A) > 0 for some n > 1.

liv) => (i). Suppose В eana T~XB = B. If 0 < m{B) < 1 then 0 = m(B n (A B)) = m(T""B о (A B)} for all n > 1, which contradicts (iv). □

Remark. Noiice that we could replace (iii) by the statement “For every A fc Mi \wih /«(.-I) > 0 and every natural number N we have m({j,‘ v T~"Ai = I" because y,; s T ~"A - T Л(ЫИ‘_„ T~"A). Consequently we could replace

2S

I Measure-Preserving Transl'ormmions

(iv)b\ ".W even .-1, 4> c-.//’ with iu(.!) > 0 ,m(i>) > 0 and every natural number .V there cm.sis n > N willi m(T~"A nli) > O'. One can think of (iii) and (iv) as sayinu that the orbit ''!' "A () of any non-trivia! set A sweeps out the и hole sp ice A (or that each non-trivial set A has a dense orbit in a measure-theoretical sense).

The next theorem characterises ergodicity in terms of the operator UT.

Theorem 1.6. If (A\.W,in) is a probability space and T:X -» X is measure-preseri ini) then the followiiuj statements are equivalent:

(i) T is eriioilic.

(ii) Whenever f is measinable and (/ ^ T)(x) = f(x) Ухе X then f is constant a.c.

(iii) Whenever f is measurable and (/ T)(,v) — f{x) a.c. then f is constant

a.e

(iv) Whenever f e L2(in) and ( f о T){x) = f(x) Va 6 X then f is constant a.e.

(v) Whenever f e L2(m) and (/ - T)(x) = f(x) a.e. then f is constant a.e.

Proof. Trivially we have (iii) => (ii), (ii) => (iv), (v) => (iv), and (iii) => (v). So it remains to show (i) => (iii) and (iv) => (i). We first show (i) => (iii). Let T be ergodic and suppose / is measurable and f ° T = / a.e. We can assume that / is real-valued for if / is complex-valued we can consider the real and imaginary parts separately. Define, for к e Z and n> 0,

A'(/<,») = {.v:k/2" < J\x) < (k + l)/2"; = f~l{[k/2", (k + l)/2")).

We have

I 1 X(k,n) Л X(k,n) c {x:(f о T)(x) Ф f(x)}

and hence m(T~ lX(k,n) Д X(kjufc) = 0 so that by (ii) of Theorem 1.5 m(X{k,n)) — 0 or 1.

For each fixed n UkeZ A"(/<, н) = X is a disjoint union and so there exists a unique k„ wilh m(X(k„,n)) — 1. Let Y = ff^=i X(k„,n). Then m(Y) = 1 and / is constant on Y so that / is constant a.c.

(iv) => (i). Suppose T 1E = E, E e Then yE e L2(m) and ('/E ° T)(x) -X,;( v) Vv e X so, by (iv), yv is constant a.c. Hcnce yE — 0 a.e. or yE = 1 a.e. and m(E) = \ '/j. dm — 0 or 1. □

Remarks

(1) A similar characterization in terms of Z/(m) functions (for any p > 1) is true, since in ihc last part of the proof yE is in L‘\m) as well as L2(m). Also we could use real iJ’Un) spaces.

$1.5 Ergodicity

29

(2) Another characterization of ergodicity of T is- whenever /:X -* R is measurable anJ f(Tx) > f(x) a.e. then / is constant a.e.

This is clearly a stronger statement than (iii). To see that ergodicity implies the staled property, let T be ergodic and suppose f(Tx) > f(x) a.e. If / is not constant a.e. then there is some r e R with В = (x e А^Дх) > c} having

0 < m(B) < 1. But T~lB => В so m(T~lB ДВ) = 0 and hence m{B) = 0 or

1. This contradicts 0 < m(B) < 1.

We want to analyse our examples to see which of them are ergodic. The next result will be useful for this and it also relates the idea of measure-theoretic dense orbits (Theorem l.S(iii) and (iv)) to the usual notion of dense orbits for continuous maps. We shall say more about this in Chapter 6. Recall that the ст-algebra of Borel subsets of a topological space is the ст-algcbra generated by the open sets.

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