# An introduction to ergodic theory - Walters P.

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By Lemma 8.5 choose a partition ^ = {At,... ,Ak} of (Х,ЩХ)) so that diam(4j)<£ and ц(сА,)=0 for 1 </<k. Then H„j\/?=o T~‘£)—log^fc, X) since no member of \/?=o T~‘£ can contain more than one member of En and so s„(e,X) members of \/;=o each have <T„-measure \/s„(e, X) and ' the others have c^-measure zero. Fix natural numbers </, n with 1 < q < и and, as in Remark (2), define a(j), for 0 <7 < q — 1, by a(j) = [(/1 —j)/</].

190

8 Rel-iiionship Between Topological Entropy and Mca*ure-Thuaretic Entropy

Fix 0 < j < q — 1. From Remark 2(ii) wc have

V?=0 Т~Ч = V?=V 7",f, + '’ V?=o T-'Z V Vie6 T~l4 and S has cardinality at most 2q Therefore

\ogsn(e,X) = hJ^ T-'tj

mj)-l / q- 1 \

< I Нвя(7’-‘'*+л V T-'cW V ЯП„(Т-^)

r=0 \ 1=0 / *eS

by Theorem 4.3fv iii)

лJ)-1 /и-i \

< Z He+T-ri'A V T’”, )+ 2qlog(/i) by Corollary 4.2.

r=0 \l = 0 /

Sum this inequal ty over j from 0 to q — 1 and use Remark 2(iii) lo get qlogsJfcX)*^ Т-‘А + 2/1оg(k).

p= о \; = o /

If we divide by n and use Remark (1) we get

- logi„(£, X) < HVn fV T~lz \ + 2-^ log(A). (*)

П \i — 0 J n

By Remark 3 we know the members of \J\Zl T~have boundaries of ^-measure zero, so limjJ(n unj(B) = ц(Б) for each member В of \'(=l T~‘£, (Remark 3 §6.1) and therefore lim^ Н^{\JtZl T~l£) = Я„(\/7=о Т~‘£). Therefore replacing n by n} in (*) and letting j go to со we have qs(c, X. T) < HjSfIZо T~‘£). We can divide by q and let q go to oo to get s(e,X, T) < h„{T,Z) < /ifT),. □

Corollary 8.6.1. Let T:X -* X be a continuous map of a compact metric spa^e. Then

(i) h(T) = sup{^(T)|^ £ £(X,T)}.

(ii) h(T) = h(T\nm).

(iii) h(T) = h(T\ ft rx).

*-o

(iv) If, for i = 1,2. T,: X, -» X,- is a continuous map of a compact metric space and if there is a bijection ф:Х1 —> X2 which is bimeasurahle (i.e. ф and ф~1 are measurable) and фТ1 = Т2ф then h(Tx) = h(T2). (This generalises the fact that topological entropy is an invariant of topological conjugacy.)

Proof

(i) Let e > 0 be given. Choose ц e M(X, T) such that ',-(r)>tl/e = со.

£8.3 Measures with Maximal Entropy

191

fax.T)m dr(m) is the ergodic decomposition of/i then, by Theorem 8.4, hJT) = jE(x.r)^m(T)dr(m) and so hJT) > hJ^Tj — e. for some m e E(X, T). Hence

i ,t)^SHT]~2e if,'(r)<3C

U ' jl/e-Е tfUJT) = oc

so that sup{/j„|T)]m e E(X, Г)} = h(T).

(ii) By Theorem 6.15 we have u(Q(T)) = 1 V/; e V/( A. T) so th^t

sup(/7„(r)|/i e M(X, T)} = sup'/iJ((7')|/< e Л/(0(7"), T|n,n)}.

(iii) If цеМ(Х.Т) then ц(ТгХ) = ц(Т'пТ''Х) = fi{X) = 1. Therefore МПп*= i TnX) = 1 VfieM(IJ) so that we can identify ,\/(A, 7 ) and M(| lo TnX,T). The result follows from the variational principle.

(iv) We have i.i&M(Xl,T1) iff ц ф~1 e A/(A\, T2). Also hll(Tl) = кц.ф-АТг) so the result follows from the variational principle. □

§8.3 Measures with Maximal Entropy

The variational principle gives a natural way to pick out seme members of M(X,T).

Definition 8.2. Let T:X -* X be a continuous transformation on a compact metric space X. A member p. of M(A, T) is callcd a measure of maximal entropy for T if /iM(T) = h(T)

Let MmaJ X, T) denote the collection of all measures with maximal entropy for T. After the next theorem, which gives the properties of A, T), we shall give an example where Mmax(A, T) is empty.

Theorem 8.7. Let T: X-* X be a continuous transformation of a compact metrisable space. Then

(i) Mmax(A, T) is convex

(ii) If h(T) < oc the extreme points of WJA', T) are precisely the ergodic members of Mmax(X, T).

(iii) If h{T) < со and Mmax(X, T) ф 0 then Mraax(A\ T) contains an ergodic measure.

liv) If h(T) = со then Mmax(А, Г) # 0.

(v) If the entropy map is upper semi continuous then MmaJX, T) is compact and non-empty.

Proof

(i) This follows since the entropy map is affine (Theorem 8.1).

(ii) If ц e A/max(X, T) is ergodic then it is an extreme point of M(X, T) (Theorem 6.10(iii)) and hence of A/mjx(X, T). Now suppose ц e A/max(A, T)

192

8 Relationship Between Topological Entropy jnd Measure Theoretic Entropy

is an extreme point of T) and p = ppx + (1 — p)/j2 for some p e [0,1],

pup2 e M(X, T). Then, since h(T) = h„(T) = phPl(T) + (1 - p)h,JT) (Theorem 8.1) and htli(T), 1гЦ2(Т) < h(T) (Theorem 8.6). we must have pt. p2 e Mmail(X, T). Hence p = p: = p2 and p is an extreme point of M(X, T) and hence ergodic (Theorem 6.10).

(iii) Let p e Mmx(X, T) and let p = JE(x,n »uh(m) be the ergodic decomposition of p. By Theorem 8.4 li(T) = ЛДГ) = \FAXT)liJT)di(m) and since hjT) < h(T) (Theorem 8.6) we have m e Мтлх(Х, T) for т-almost all m.

(iv) By Theorem f hoose p„ e M(X, T) with h^JfT) > 2”. Let

.« j

^' = Z ^ИлеМ{Х,Т).

n= 1 z

Since

N 1 1

V = L y, V-* + v for some v 6 MX, T)

и ® 1 —

we have

N ,

ЙДТ) > £ — ft^fT) > N for each N (Theorem 8.1).

n = 1 “

Hence p e Мтях(Х, T).

(v) The set Mmax(X, T) is non-empty because an upper semi-continuous function on a compact space attains its supremum. The uppei semi continuity also implies Mmax(X, T) is compact because if p„ e Mmax(X, T) and p„ -* p e M(X, T) then ЛДТ)> limsupn^„,/i(J7') = h(T) so that pe .Мтлх(ЛГ, T) □

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