# An introduction to ergodic theory - Walters P.

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h^(S) = J£(i. S) hp(S) dx ф~ \p) = J£(jf T> l4m(S)dT(m).

Since % = ф~Ч], where >] is the natural generator of S, we have hj,m(S) = li4,m(S,4) = hm(T,t) Vm e ЩХ, T). Therefore

hu(T, c) = Jzavn hJT^)dT(m)

(ii) Choose finite partitions £q, в > 1, of (X, ЩХ)) with + i for all

cj and di< m(c,) -* 0. Then lim^x hJT,^4) = lijT) Vm e M(X, T), by Theorem 8.3, so by the monotone convergence theorem for the measure t we have

hJT)= lim hM(T,Q = lim Г h„(T,Qdi(m)

q~* oo q~+ сю " ' * *

= L.rA'<T>'hw- n

§8.2 The Variational Principle

In this section we prove the basic relationship between topological entropy and measure-theoretic entropy: if X is a continuous map of a compact motric space then h(T) = sup {ht,(T)\ti e M(X, T)}. The inequality sup{/i(1(T)|/i e M(X,T)} < h(T) was proved by L. W. Goodwyn in 1968. In j970 E. I. Dinaburg proved equality when X has finite covering dimension and later in 1970 T. N. T. Goodman proved equality in the general case. The elegant proof we present is due to M. Misiurewicz.

Wc shall need the following simple lemma, where we use с В to denote the boundary of a set B(PB = B\ini(B)).

Lemma 8.5. Let X be a compact metric spine and fi e M(X).

(i/ If x e X and 3 > 0 there exists 3' < 3 such that /.i(3B(x; <>')) = 0.

(ii) If б > 0 there is a finite partition £ = {Au ..., Ak] of (X .A A')) such that diam(/lJ) < 3 and n(dAj) — 0 for each j.

188

8 Relationship Between Topological Entropy and Measure- Theoretic Entropy

Proof

(i) This is clear since we cannot have an uncountable collection of disjoint sets of positive measure.

(ii) By (i) there is a finite open cover fi *■ {Bi,... ,B,} of ,Y by bulis of radius less than 6/2 wkh ц(с B() = 0 for all j. Let .4, = B, and for n > 1 let A„ = tin (Z3! u B2<u u B„_Д Then £ = {A ,, ..., Ar\ is a partition of (X, Jd(X)), diam(/i„) < 8, and >ince dA„ c= Q"=, dBt we have ц[дА„) = 0 for all n □

We now collect together some results we will use in the proof of the variational principle. In this sccnon X will always denote a compact metric space and ЩХ) the ст-algebra of Borel sublets.

Remarks

(1) If ^ e M(X). 1 < / < n. and />; > О, X?*i Pi ~ * l^en

Hf»K) ^ t P.HJO i = 1

for any finite partition с of(A,.^(A)). (For the proof see the remark following Theorem 8 1.)

(2) Suppose q, n are natural numbers and 1 < q < » For 0 < j < q — 1 put a[j) — [(»i — j),'q\. Here [fc] denotes the integer part of b > 0. Wc have the following facts.

Ii) «(0) > «(1) > ■ • • > a(q - 1)

(n) Fix 0 < j < q — 1Л hen

{0,1,. .., n — 1} = {j + rq + /|0 < r < a(j) — 1,0 < /' < q — 1} u S

where

S = {0,1,... ,j - 1 ,j + a(j)q, j + a(j)q + 1.....n - 1}.

Since j + a(j)q > j 4- [((n —j)/q) — l]q = n — q, we have the cardinality of S is at most 2q.

(iii) For, each 0 <j < q - 1, (a(j) - l)q +j < [((« - j)/q) - \]q +j = n — q. The numbers {j + rq\0 <j<q—l,0<r< a( j) — 1} are all distinct and are all no greater than n — q.

(3) If ц e M(X, T) and if ц(дА,) = 0, 0 < / < n — 1, then

,(a(n т'ч)) = 0 since8("П T~lA^j^ U T-ldAt.

Theorem 8.6. Let T\X —* X be a continuous map of a compact metric space X. Then h(T) = sup{^(T)|// e M(X, T)}.

58.2 The Variational Principle

189

Proof

(1) Let ц e M(X, T). We show in this part that hJ,T) < h(T). Let £ = {Ab ... ,.4k] oe a finite partition of (Х.Л(Х)). Choose e > 0 so that e < l/(k\ogk}. Since ц is regular there exist compact sets Bj cr AJt 1 Cj & k, with p(A/ Bj) < e. Let be the partition t] = {B0. B,,... ,Bk) where B0 = A' Bj. VVe have /*Ш0) < kc. and к к

since if i Ф 0.--n— = 0 or .

/<(£,)

< ц(В0) \og(k) by Corollary 4.2.

< krAogik} < 1.

The reason we can bring in topological entropy is that for each i / 0, B0 и В, = В} is an op^n set so /? = \B0 u . , B0 u is an open

cover of A. VVe have, if n > 1, HtД/|‘=о T~‘ij) < log Л'(\/^ о T by Corollary 4.2, where N(\J"Zo T~lt]) denotes the number of non-empty sets in the partition so Я„(\/7=о Г"1!/) < log(A'(V-~o T~'P) ■ 2").

Therefore

hJLT.ij) < h{T,p) + log2 < h(T) + log2,

so

h„(T, £) < ИЦ(Т, r]) + H,№/>]) by Theorem 4.12(iv)

< /i(T) + log2 + 1.

This gives ЙДТ < h(T) + log2 + 1 for any continuous map T with e M(X, T). It therefore holds for T" so nliJT) < nh(T) + log2 + I by Theorems 4.13(i) and 7.10(i). Hence hЦ(Т) < h(T).

(2) Let e > 0 be given. We shall find some ц e M(X, T) with hJT) > s(e, X, T), and this clearly implies sup {hJT)\u e W(A\ T)} > h(T).

Let E„ be a \n,z) separated set for X of cardinality s„(t, A'). Let a„ e M(X) be the atomic measure concentrated uniformly on the points of E„ i.e. ffn = (l/s^ ЛГ))Х*еь-„^- Let /j„ £ M(X) be defined by ^и = (1/«) У]Г=о 1 7'~‘-Since M(X) is compact we can choose a subsequence of natural numbers such that ПгП;_л (l/и,)logsnj(£, A) = s(t:,X. T) and {ц,,} converges in M(A) to some ц e M(X). By Theorem 6.9 we know ^ e M(X, T). We shall show

h,(T)>S(E,X, T).

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