# An introduction to ergodic theory - Walters P.

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the two-sided shift. Then T has a unique measure with maximal entropy and this unique measure is the (l/к, 1 /к,..., \/k)-product measure.

Proof. We know /i(T) = logk. Suppose htl(T) = log Ac. Let £ = {/10,... ,Ak_,} be the natural generator (i e. Aj = , x0 = j}). Then log A: = /;„(T) <

(1 /n)HjSJ1Zo T~'E) < (l/n)logk'‘ = log к by Theorem 4.10 and Corollary 4.2.1.

Therefore HJ\/"=o T~lc) — log kn so, by Corollary 4 2.1 each member of \J "Jj* 7 has measure (l/к"). Hence ц is the (l/k,, l/k)-product measure. □

This was generalised to the case of topological Markov chains by Parry. Recall from §6.6 that if T: XA -* XA is a two-sided topological Markov chain and A is an irreducible matrix then there is a canonically defined Markov measure given by a probability vector (p0,... ,pk_ t) and stochastic matrix () as follows. If Я is the largest positive eigenvalue of A and (n0,...., uk. is a strictly positive left eigenvector and (r0,. .. ,^-j) is a strictly posithe right eigenvector with ]Г?=о ы.-и,- = 1 then pt = и{ь{ and pu = a^vp.v,. We call this measure the Parry measure for T:XA-> XA.

Theorem 8.10. If T:XA -* XA is a two-sided topological Markov chain, where A is an irreducible matrix, then the Parry measure is the unique measure with maximal entropy for T.

Proof. We know h{T) = log Я by Theorem 7.13.

Let ц denote the Parry measure. We first show p e Mmax(XA, T) by showing hJiT) = к^Я. By the formula for the entropy of a Markov measure

§8.3 Measures vith Maximal Entropv

195

(Theorem 4.27) we have k-l

4T)=~ Z Wil,^Iog№ i.j= 0 / li \ AVi .

fc- l

= - Z

uie,tjLj

i, j = 0

pOgUy + logl’j - log;.'- log L,f]

1 k-l

= 0 - Z uivi log L'j + l°g^ + Z ад l°g ^ since u.j6 {0,1} j=0 i=0

= log/..

We now show ^ is the only measure with maximal entropy. We know // is ergodic since the matri" (cii;) and hence (p;j) is irreducible. By Theorem 8.2 and 8.7(v) we know that if {/*} Ф Mmix(XA, T) there is another ergodic member m of Afma!C(A'M, T). By Theorem 6.10(iv) m and /t are mutually singular so IE e ЩХл) w,th ц(Е) = С and m(E) = 1.

Let <!; = {Л0,...,/4к_denote the natural generator i.e.

Aj= c-.Y (|A'0 =./}.

Since j/(Vr= n T~li) * %{XA) we can choose E„ e t/(V?=-iп-n T~'£) with (hi + /<)(£„ A E) -» 0. Hence ц(Еп) 0 and m{E„) -* 1.

Let i]n denote the partition ij„ = {£„, A ',£„}. Then

.eg. - ;ur> s ^ B.(V r-{) - V „ r-‘i)

i

[ — m(£„) log(£n) - (1 - /«(£„)) log(l - )»(£„))

~2n- 1

+ m(E„)log0n(E„) + (1 - /«(£„))log0„(X\En)]

where 0„(B) denotes the number of elements of V/7= -m- n T~'£ that intersect В £ j*(V"=-(„-i) T~'q). (Here we have used Corollary 4 2 to estimate the entropy of the partitions of the sets £„, X\E„ induccd by

Therefore

'-log/. <

1

2n — 1

V

;= -in- и

/E.„ 6ДЕ.) , „ ,r„1 , 0XX\En) w(£n)log—+ (1 - ni(£„))log-

’m(En) v "" °l-m(£„)

However if С £ \/?=-(я- n T~l£, say

C {{Xn} - OO I (x-(n- 1) Xn - l) 0" —(n— 1)’ • • • »Уи- l)j

(*)

196

8 Relationship Between Topological Entropy and Measure-Theoretic Entropy

then

So if

then

р=-(я-1)

- —since “w,.. “ L

a = min' UjUj, b = max u,r,-

05i. j£k- 1 05 i. i2»*~ 1

T herefore

^<H(C)<-VC e V Т-г ^ 0n(B) < pi(B) < b -S®, VBe.c/1 V ^'Л

* Л \i=>-(n-l) /

Using this with В = E„ and В = X\En in equality (*) gives

l0sis2^b[",(£-|l08(“fi&r)

+ (1 - m{En))\og

and hence

ME,

0- fi(En))l2n-1\

a(l - m(En)) J

0 < m(En) log

Щ

En)J

\am(En

+ (1 - w(£:„))log(l - pi(E„)) - (1 - m(E„))log(a(\ - m(E„))).

When n tends to oo the limiti of the three right-hand terms are — ос, 0, 0 respectively. This contradiction shows that Мтлх(ХА, T) = {pi}. □

Remark. If T:X -* X has a unique measure, pi, with maximal entropy then one would expect pi to be important because it w as characterised in a natural way from the variatior principle. We shall generalise the varsational principle in Chapter 9 and this allows us to characterise other measures in a similar way.

§8.4 Entropy of Affine Transformations

In this section we study the relationship between topological entropy and the Haar measure entropy of an atfine transformation. We also calculate these entropies for an affine transformation of an п-torus. The first result shows Haar measure is a member of Mmax(G, T) when T:G -* G is an affine transformation of a compact metric group.

;v4 E'ntropy of Affine Transformations

147

T heorem 8.11. Let G he a compact metrisable (/тир and 7 a • I an affine transformation of G. Li t m denote (normalised) Haar measure on G. I lien hm{T) = hm(A) = Л(Ч) = ЫТ). It d denotes a leh-inrariant metr,c on G then h{TI = limf_0 limsupn^ , —(1 n) log;»!((")'' .,1 -1 ~'B(e: <:)). where e denotes the identity element of G and Hie', r.) is the open hall centre с and radius r. wlih respect to the metric d. (This limit clearly exists or is r .)

Proof. By Thco.crn 8.6 wc have hJT) < /i(T). Suppose d is a lefi invariant metric on G. Put D„(x,s, T) — P|"-n T~kB{Tkx: >:). By induction wc shall show that T~kff(T*x: c) = x ■ (A kB(e:i:)). h is true for к = 0 b\ the invariance of the metric d. Assuming it holds for к we prove it for к + 1:

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