# An introduction to ergodic theory - Walters P.

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The next result expresses the mixing concepts in functional form. This will be useful for checking whether examples have the mixing properties. Recall that UT is defined on functions by UTf = f ° T.

Theorem 1.23. Suppose (X,36,m) is a probability space and T.X-+X is measure-preserving.

(i) The following are equivalent:

(1) T is ergodic.

(2) For all f, g e L2{m) limn^00(l/n) Y!!=o (^r/.0) = (/. l)d.0)-

(3) For all f e L2(m) £“:0‘ (VlTf,f) = (/, 1)(1,Л

(ii) The following are equivalent:

(1) T is weak-mixing.

(2) For all f,g e L2(m) lim^d/n) <5 |(U‘Tf,g) - (/, l)(l,g)| = 0.

(3) For all f e L2(m) lim^, (1/л) %=о WrfJ) ~ (/. 1)(1,/)| = 0.

(4) For allfe L2(m) lim_ (1/и) £?-<} \{U'rfJ) - (/, 1)(1,/)|2 = 0.

(iii) The following are equivalent:

(1) T is strong-mixing.

(2) For all f, g e L2(m), lim„_., (U"Tf,g) = (/, l)(l,ff).

(3) For all f 6 L2(m), Um^m(U"Tf,f) = {f 1)(1,/).

46

1 Measure-Preserving Transformations

Proof, (i), (ii) and (iii) are proved using similar methods. We shall prove (iii) to illustrate the ideas. Slight modification of this proof will prove (i) and (ii).

(2) => (1). This follows by putting / = /и, g = %D, for A, В e £fi.

(1) => (3). We easily get that for any A, Be SI, (U"TzA,xB) — (xA, 1)(1,j6h)-Fixing B, we get that (C/"r/i,Xc) -* (h, 1)(1 ,/в) f°r апУ simple function h. Then, fixing h, wc get lhat (U"rh, h) -► (h, l)(l,/i). So (3) is true for all simple functions.

Suppose / e L2(m), and let e < 0. Choose a simple function h with Ц/ — /i||2 < £, and choose N(r) so that n > N(e) implies

|(LWi)-(M)(1,/i)|<£.

Then if n > N(c)

\wfj) - (/. m,/)\ < W'Tfj) - (unThj)\ +1 (1/-АЛ

- (Ujh,h)\ + |(U”Th,h) - (/|,1)(1,Л)| + \(h, 1)(1 ,A) -(/,l)(l,A)| + |(/,l)(l,/i)-(/,l)(l,/)|

< |(W - h),f)| + \(UnTh,f - h)|

+ £ + |(1,/01 |(й - /, 1)1 + |(/, 1)11(1, л - f)\

< ii/ - Л1Ы1/И2+11/ — ^luiiAiu+£+mw - ли2

+ ||/||2||й — f\\i by the Schwartz inequality ^ e||/||2 + £(||/||2 + £) f £ + (Ц/Ц2 + £)£ + 4fh-

Therefore lim„-„(С/?/,/) = (/, 1)(1,/).

(3) => (2). Let / e L2(m) and let Ж; denote the smallest (closed) subspace of L2{m) containing / and the constant functions and satisfying UTyff а Ж{. The set

3Ff = ^ge L2(m): lim (U"Tf,g) = (/, l)(l,gf)j

is a closed subspace of L2(m) and contains / and the constant functions. Since it is UT invariant it contains Ж s. Ifg e Ж) then (V'Tf,g) = 0 for n > 0 and (l.gf) = 0 and therefore Ж^ a iF f. Hence &s = L2[m). □

Remark. Another form of weak-mixing is in terms of sets of density zero. Let us suppose that (X,9S,m) has a countable basis. Then T is weak-mixing iff there exists a subset J of Z+ of density zero such that lim =

(/>1)(1>S0 for all/,geL2(m).

The next result connects weak-mixing of T with the ergodicity of T x. T.

Iheorem 1.24. If T is a measure-preserving transformation on a probability space (X, then the following are equivalent:

(i) T is weak-mixing.

(ii) T x T is ergodic.

(iii) T x T is weak-mixing.

JJI.7 Mixing

47

Proof. We first show (i) =?• (iii). Let A, B,C,De bS. There exist subsets Jt, J2 of Z* of density zero such that

lim m(T'nA n B) = m(A)m(B)

J j $ л-* oo

lim m(T nC n D) = m(C)m(D).

00

Then

lim (m x m){(T x T) "(>4 x С) n (B x £>)}

lim ™(T"M n B)m(T~"C n £>)

J 1 wJz ^ Л—*• CO

= m(/l)m(B)w(C)m(D)

= (m x ш)(у4 x C)(m x т)(В x £>).

By Theorem 1.20 we know lim„_.00(l/#j)£"r(J |(m x m){[T x T)~"(A x C) n (В x £>)} — (m x m)(A x C)(m x m)[B x £>)| = 0. Since the measurable rectangles form a semi-algebra that generates 88 x sg Theorem 1.17 asserts that T x T is weak-mixing.

It is clear that (iii) implies (ii).

We now show (n) =?• (i). Let A,Be&. We shall show limn-,„ (l/и) £"=o (m(T'lA n B) — m(A)m(B))2 = 0. We have

1 П— 1 1 л — 1

- £ nB) = -[(mx m)((T x T)~‘(A x X) n (В x X))

n i = o n f=0

-*(m x m)(.A x ЛГ)(/и x m)(B x X) by (ii)

= m(A)m(B).

Also

J И—1 | И“1

- £ (т(Г_‘Л n В))2 = - £ (m x m)((T x T)_i(/1 x /1) о (B x B)) n i=0 и 1=0

-»(m x m)(A x Л)(/п x ш)(В x B) by (ii)

= т(А)гт(В)2.

Thus **-

1 "-1

- £ {m(T~‘A nB)- m(A)m(B)}2

n

1 = 0

1 n_1

= - £ {ш(Т_‘/4 n B)2 - 2m(T~lA n B)m(A)m(B) + m(A)2m(B)2} ” i = 0

П “c,

-» 2m(A)2m(B)2 - 2m(A)2m(B)2 = 0.

■Therefore T is weak-mixing by Theorem 1.21. □

48 l м ensure-Preserving Transl'ormalions

Remark. It is easy ю show that 7 is strong-mixing iff T x T is strong-mixing.

We now relate the weak-mixing of T to a spectral proDcrty of the operator UT on L2(m).

Definition 1.7. Let The a measure-preserving transformation of the probability space (X,3$,in). We bay a complex number Я is an eigenvalue of T if it is an eigenvalue of the isomctry UT:L2{m) -» L2(ni) i.e. if there exists f e L2(m) f Ф 0 with UTf = Я/ (or f{Tx) = /.f(x) a.e.). Such an / is called an eigenfunction corresponding to Я.

Remarks

(i) If Я is an eigenvalue of T then |Я| = 1 since

ll/ll2 = Prf\\2 = (UTf. UTf) = WAA/) = W2||/||2.

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