# An introduction to ergodic theory - Walters P.

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„ p(B n E) u(Bn(X\E))

p(E) p(WE)

Note that p} and p2 are >n M(X, T), /z, ф p2, an£l

p(B) = /i(£)/i,(B) + П - H<E))p2(B).

Therefore p is not an extreme point ol M(X, T).

Conversely, suppose p e M(X, T) is ergodic, and

P = РР i + (1 - P)P 2

where plt p2 e M(X, T) and p e [0,1]. We must show = p2. Clearly /ii « p (pi is absolutely continuous with respect to p) so that the Radon-Nikodym derivative dpjdp exists, (i.e.,

dp Ax)

i(E) = J£ dp{x), V£ e ЩХ).

dp

See Theorem 0.10). We have dp^fdp > 0. Let

56.2 Invariant Measures for Continuous Transformations

153

Wc have

f , ^-d^ + Г ^ic^ = /(1(£) = /t1(T'-1£)

=Tf f ^-J/«

jEr.r-iE dn h -'F'F dfl

ao that

dfh , Г

Г ~dfi = Г d>‘-

Jr\T-«£ С/ц Л Jr->£ £ </,,

c//t J7' ‘L f dfi

Since d^i/dn < 1 on £\T_I£ and <//<,iclji > 1 on T~lE,E and since ц(Т~'Е\Е) = ц{Т~1Е) — fi(T~ 1EnE)= ц(Е) -ц(Т~1Егл Г) = nft\T~lE) we have ц(Е\Т~1Е) = 0 = ;t(7'~1£\£). Therefore ^(7'_1£Д£) = 0 so ^(E) = 0 or 1. If //(£) = 1 then /^(.V) = \E(dfiljdfi)dfi < ц(Е) = 1 contradicting ц}(Х) — 1. Hence we must have ;i(E) = 0.

Similarly if F = {.v|dftjdfi > 1} we have //(£) = 0 so that dfi1/dfi = 1 a.e. (fx). Hence fil — fi and so /j is an extreme point of M(X, T).

(iv) By the Lebesgue decomposition (Theorem 0.11) there are unique probability measures ar>d a unique p e [0,1] such that fj. = p/t, + (1 — p)fi2 where /г, « m and //2 is singular with respect to m. But ft = Тц = pTjt, + (1 — р)Тц2 and since Т« 7 m = m and Tfi2 is singular with respect to Tm = m the uniqueness of the decomposition imply //,, ц2 e M(X, T). Since /i is an extreme point we must have either p = 0 or p — 1.

If p = 0 then ц = Цг and so ц is singular relative to m. If p = 1 then // « m and we can argue with dfi/dm as in (iii) to get ft = m, a contradict.on. □

Remarks

(1) The second part of the proof of (iii) shows that if /^,/ге M(X,T), fil « /I and /л is ergodic then /i, — ц.

(2) Since M(X, T) is a compact convex set we can use the Choquet representation theorem to express each member of M{ X, T) in terms of the ergodic members of M{X, T). If E(X, T) denotes the set of extreme points of M(X, T) then for each ц e M(X, Г) there is a unique measure r on the Borel subsets of the compact metrisable space M(X, T) such that z(E(X, T)) = 1 and V/ 6 C(Af)

jxf(x)dll(x) = Ji;(v r) ^/(jcJr/mCx^rM.

We write ft = \е{х.т),п^т(п1) and call this the ergodic decomposition of ц. See Phelps [1]. Hence every ц e M(X, Г) is a generalised convex combination of ergodic measures. This is related to the decomposition of a measure-preserving transformation into ergodic transformations (see §1.5).

We shall now interpret ergodicity and the mixing conditions in terms of the weak*-topology on M(X, T).

154

6 Invariant Measures for Continuous Tr.msform.itions

§6.3 Interpretation of Ergodicity and Mixing

Let T:X -* X be a continuous transformation of a compact metric space. We say ц e \1{X, T) is ergodic or weak-mixing or strong mi.ving if the measure-preserving transformation T of the iheasure space (X, Л(Х),/i) has the corresponding property.

Recall that is ergodic iff V/,</ e L2(j«)

- Z Г/(7'л)б|(л')^(л)-» (fdfi fg dfi.

W ft ,--q v

Ve change this slightly for our needs.

Lemma 6.11. Let ц e M(X, T). Then

(i) fj. is ergodic if V/ e C(X) Vg e L'(;i)

“ X Jfdft Jffdfi-

(ii) /.i is strong mixing iff V/ e C(X) Vg 6 Ll((i)

^f{Tix)g(\)dii(x) — ^fdfi Jg d[i.

(iii) /j ts weak mixing iff there is a set J of natural numbers of density zero such that V/ e C(X) 7<y e L’(//|

^ lim §f[T‘x)g{x)dn(x)-*§fdn Jgdfi.

Proof

(i) Suppose the convergence condition holds and let F, G e L2{fi). Then

G 6 L4rf so i У J/(T‘x)G(x) J/4x) — Jyrf/i jGrf/i V/ 6 C(X).

Now approximate F in L2(fx) by continuous functions to get

“ Z §F{T!x)G(x)dn{x)-> §Fdn jGfa

Now suppose ц is ergodic. Let / e C(A'). Then / e L2(^) so if h e L2{//) we have

Z f/( Г‘*Ш x) Mx) -*■ J7 dn jhdfi.

If ge L*(/0 then by approximating g in Ll(n) by h e L2(/i) we obtain

“ Z JАТ‘ХМ*)Mx)dfi-

§6 3 Interpretation of Ergodicity and Mixing

155

The proofs of (ii) and (i.i) are similar and use Theorem 1.23. □

Theorem 6.12. Let The a continuous transformation of a compact metric space. Let p e ЩХ, T).

(i) ц h prgodic iff whenever m e M(X) and m « p then

1 " 1

- У f'm -> p. n “o

(ii) p is strong mixing iff whenever m e M(.V) and in « p then Tnm —> p.

(iii) u is weak mixing iff there exists a set J of natural numbers of density zero such that whenever m e M(X) and m « p then lim_,)n-OD Tnm -* p.

Proof

(i) We use the ergodicity condition of Lemma 6.11.

Let p be ergodic and suppose m « p, m e ЩХ). Let g = dm ‘dp e Lx(p). If / 6 C(X) then

(W1 I T'ni\ = - X jV° T‘ dm = ~ X \f(T'x)g(x)dp(x)

J \n 1 = 0 / 4 i-0 J n 1 = 0 J

->■ jfdp^gdp = jf dp J1 dm = ^f dp.

Therefore (l/n)X"=o T m -> p.

We now show the converse.

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