# An introduction to ergodic theory - Walters P.

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(v T-‘ir) + fsjdvz X KC){-log^C) + a(C)]

'■i“° ' Cf’vn

1*0

<, log Y e“<C) by Lemma 9.9.

CeV Т"ч 1‘ 0

For each Сe\/;=о T~'r\ choose some xeCso that (S„/)(x) = a(C). Since £ is (n,6) spanning choose y(C) e £ with d(T'x, Tly{C)) ^ 5, 0 <, j < n — 1.

1)9.3 The Variational Principle

219

Then a(C) < (S„/)( WC)') + nг.. Also each ball of radius 6 meets the closures of at most two members of»/ so if у e E then ]C 6 \/7=o 7"’,'/|,WC) = >■} has cardinality at most 2n. Therefore

£ ^to-nt ^ <^2n Y. e<s"/'(>)

• » • » ,. - L

Cc V ctVr1!

and so

<- о i-о

log I Y e“(C>\ — пе< n log 2 + log Y eiS”/Hy>

[Cc V T-'4

yeE

Hence

1«,(V 7-',) + J/* - i H.( V гл) + ; Js./*

< e + log 2 + - log £ e(SF,/Wv>

n

y-E

^ e + log 2 + - log P„(T,/, <5)

n

and therefore

ЛН(Т,»/) + J/ J/; ^ e + log 2 + P(T,f, S)

< e + log2 + P(T,f).

Now hlt(T,£)<hlt(T,ri) +(Theorem 4.12(iv)) so that hlt(T.l,) + J/ dfi<2a + log2 -t- P(T,f) and hence h^T) + dn<,2a + log2 + P(T,f). This holds for all continuous maps T and all / e C(X. R) so we can apply it to T" and S„f = Y"^o f° Tl to get и[АДТ) + J/^] ^ 2a + log 2 + nP(T,f) (by Theorem 9.8(i)). Since this holds for all n we have

hhm + Jfdn<P(T,f).

(2) Let e > 0. We shall find ц e M{X, T) with й„(Г) + \/<1ц > P(T,f,e), and this clearly implies sup{^(T) + {/e ЛДХ, T)} ^ P(T,f).

Let E„ be an (n, e) separated set with

log Y e(S"/,,y);> logРЯ(Г,/, e)-1.

Let <rB e M(X) be the atomic measure concentrated on E„ by the formula

o. =

Y e^fm5y

yeE„

£ e(S„/)d) xc£„

Let e M(X) be defined by

220

9 Topological Pressure and Its Relationship with Invar nt Measures

Since M(X) is compact we can choose a subsequence {n^} of the natural numbers such that

lim — log Pn,(Tj, e) - P(T,f, e) v-*> >h

and {цП]} converges in M(X) to some ц e M(X). By Theorem 6.9 we know ц e M(X, T). We shall show Л„(Т) + j/Ац > P(T,f, e).

By Lemma 8.5 choose a partition £ = {At,..., Ak} of (X,J8(X)) so that diam(/4,) < £ and ц[сА^ = 0 for 1 < i < k. Since each element of V;=o T contains at most one element of E„ we have

T~‘^j + $SJdan

= X ^({W)((Sn/Kj)-log<Tn({y}))

= log X elSnf)iy) by definition of a„ and Lemma 9.9:

ye£„

Fix natural members q, n with 1 £ q < n and as in Remark 2 of §8.2 define a(j), for 0 < j < q — 1, by a(j) = [(n - j)/q], Fix 0 £ j <, q — 1. From Remark 2(i) of §8.2 we have

и 1 o(J)-1 fl-1

V t~‘z = v т-(г«+л v t"'^v V T~lt

i*Q Г» 0 f»0 1*S

and S has cardinality at most 2<j. Therefore log £ ^«» = H0nfv (X/^„

ye£„ V/=0 / J

V 7-^+^-fV T-fcA+ fsjda,,

r = 0 \ i = 0 / \keS / J

oO)-l /«-1 \ »

^ I V 7-'£ +2qlogfc + I S„/dffn.

r = 0 \i 0 / J

Summing this over j from 0 to q — 1 and using Remark 2(iii) of §8.2 gives 4log X e(S"/K>’)£ X HcT-J V T~iA + 2q2logk + q {sjdan.

yc£n P = 0 \i = 0 /

Now divide by n and use Remark (1) of §8.2 to get

* log X e<S"/)(v> ^ + ^\ogk + q \fdtin. (*)

П У*Е„ \i = 0 / * J

Because n(dAt) = 0 all / we have by Remark 3 of §8.2 that

Hm Я (V Т~‘А = н(Ч\/ T-'A

J—CO \£«*0 / \J-=0 /

jj9.4 Pressure Determines A/(.V, T)

So replacing n by ns in {*) we get

qP(T,f,e.) £ + Щ JM<.

Dividing by q and letting q -» oo gives

P(T,f,e) Z h^T.Q + J/dM < h„{T) + J/dAi. П

Corollary 9.10.1. Let T:X -* X be a continuous map of a compact metrisable space and let f e C(X,R). Then

(i) P[T,f) = sup + J/d/i lie E(A\T)j.

(ii) Р(Т,/) = Р(Г|П(Г)./|П,Т)).

(iii) P(T,f) = Р(Т(д™,/|п ™)-

Proof. The proofs are simple generalisations of the proofs of the corresponding statements in Corollary 8.6. □

The variational principle helps to calculate the pressure of some examples. It follows readily from the variational principle (or it can be easily proved from the definition of pressure) that if T is the identity map of A' we have P( T,f) = sup/. (The ergodic invariant measures for T are the Dirac delta measures, Sx, x e X, so the formula follows from Corollary 9.10.1(i).) The following calculates P{T,f) when T is an ergodic rotation of a compact metrisable group.

Corollary 9.10.2. If T: X -» X is uniquely ergodic and M{X, T) = {m} then P(T,f) = hm(T) + \f dm.

So for a rotation T: = az of a compact metric group G with {a"} dense in С we have P(T,f) = [f dm where m is Haar measure on G

§9.4 Pressure Determines M(X, T)

We shall show how P(T, ■) determines the members of M(X,T) when T:X -* X is a continuous map of a compact metrisable space. Recall that a finite signed measureon X isamap^:j<f(_V)-»i? which iscountably additive.

4

Theorem 9.11. Let T:X -* X be a continuous map of a compact metrisable space with h(T)< oo. Let ц\Зв(Х) -* R bp a finite signed measure Then ц e M(X, T) ifflfdv P(T,f) if e CiX, R).

222

9 Topological Pressure and Its Relationship with Invariant Measures

Proof. If ц e M(X, T) then f/dpi < P^T,f) by the variational principle.

Now suppose/л is a finite signed measure and\f dpi < P(T,f)Vfe C(X,R). We first show pt takes only non-negative values. Suppose / ^ 0. If с > 0 and n > 0 we have

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