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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
Download (direct link): anintroduction1982.djvu
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Lemma 9.5. If T:X -* X is a continuous transformation of a compact met-risuble space and a is an open cover of X then for к > 0 and f e C(X,R)
lim sup - logq„(T,/, a) = lim sup 1 logq„\T,f, \/ T~'a)
n— r H n-* j H \ ;* о /
and lim„^ (l/n) logp„(T,f, a) = lim„_^ (l/n) logp„(T,f, \/‘=o T~la). If T is a homeomorphism and к, m > 0 then these formulae hold with \J^o T~‘a replaced by \/f. _m T~'cc.
Proof. One readily gets e
and
(k+ lMn'ci„+k(T,f, a) £ qn(r,f, V T-‘aj Z e<k+ :>u/Uq„+k(T,f, a)
e~(k+l},l/,,p„+k(T,f, a) £ Pn(r,f, V ^ Sk+""'"Pn+k(T,f, a).
\
The first results follow from this. To obtain the second result it suffices
to show
v-9.1 Topological Pressure
213
and
lim ^ logp„(T,f, T~ '/?) = lim ^ loj£.pn(T,f,(i)
when T is a homeomorphism and fi is on open cover of A'. Since у is a finite subcover of V"=o T~'P iff T ‘y is a finite subcover \/"=, T~l/l we have
4AT,f,T-lP)
I inf
Г. / »eC
= inf<
Z inf ps"^(v>
( e . У* Г >C
V inf
c7y «ff V
( < 7 Jtsf
inf
V is a finite subcovjer of \/ T '({)
|
... l
у is a finite subcover of \J T \fl\
jccC
я - 1
у is a finite subcover of \J T~'(l i*0
S: <Г2||Л|.
A Similar proof gives
яШТ-'п
qJLT,f,P)
so the result follows. The result for p„ follows by a similar proof. The following.result generalises Theorem 7 11.

Theorem 9.6. Let T: X —> .V be an expansive homeomorphism of a compact metric space (X,d).
(i) If a is a generator for T then
P(T,f) = lim - log/>n(T,/,oc)
n—an ft
= lim sup - logqn(T,f, a) V/ e C(X, R).
n-* a, ft
(ii) If & is an expansive constant for T then P{T,f) = P(T,/,60) = Q(T,f,S0) for all <50 < S/4 and all f e C(X, R).
Proof
(i) Let a be a generator for T. By Theorem 5.21 we have diam^ \/ T~loc^-»0
214
9 Topological Pressure and Its Relationship with Invaiiant Measures
and so by Theorem 9.4(ii)
P(T,/) = lim lim -logpjrj, \J Т-Д n- а И \ i — — it J
An application of Lemma 9.5 gives the desired result. The formula involving qn is proved in a similar way using Theorem 9.4(v) and Lemma 9.5.
(ii) Let<50 < ^/4andchoose.\.,...,.\isothat A' = (Jf-, BU',;(<V2) — 2<50). The cover a = {£(.*,-; (5/2)11 <; i <; /;} has 2d0 for a Lebesgue number so by Theorem 9.2
lim sup - log q„(T, f, a) < lim sup - log Q„( TJ. <5„)
n-* ас 91 n~* x 91
<. lim sup - log P„(T,/, <>0) ^ P{T,f).
n —* x 91
The result follows by (i) since a is a generator. □
We can use this to calculate P(T,f) when T is the two-sidtd shift on X = П-» Y' У = {Q. L — 1}’ and / depends only on the O-th coordinate i.e. /({*„}) = aXo where a0, a,.....at_, e R. If a denotes the natural
generator then qn(T,f,a) = />„(T,/,a) = (e*d + • • • + e°k ■)" so P'T,/) = log(t?"0 + • • • + e"*'1).
§9.2 Properties of Pressure
We now study the properties of P(T, ):C(X,R)~* R u {oo}. In particular we see that either P(T, •) never takes the value oo or is identically oo.
Theorem 9.7. Lei T: X -» X be a continuous transformation of a compact metrisable space X. If f,ge C(X, R), oOandc e R then the following are true.
(i) P(T,0) = h(T).
(ii) / ^ 9 implies P(T,f) <. P(T,g). In particular h(T) + inf/ P(T,f) <. h(T) + sup/.
(iii) P(T, •) is either finite valued or constantly oo.
(iv) |P(T,f,E) - P(T,g,E)| ^ Ц/ - 0||, and so if P(T, ■) < oo, \P(T,f) -P(T,g)\<\\f-g\\.
(v) P(T, -,c) is convex, and so if P(T,-)< oo then P(T. •) is convex.
(vi) P(T,f + c) = P(T,f) + c.
(vii) P(TJ + g°T-g) = P(TJ).
(viii) P(T, f + g)Z P(T,f) + P(T,g).
(ix) P(T,cf) Z cP(T,f) if с 7> 1 and P\J.tf) 7> cP(T,f) if с <. 1.
(x) |P(T,/)| £ P(T,|/|).
$9.2 Properties ol Pressure
215
Proof. Several nines in the proofs we shall use the simple inequality
sup п.- fa Д
su pb^SUPU)
when (tij), (bj) urj collections oi positive real numbers.
(1) and (ii) are clear from the definition of pressure.
(iii) From (ii) and U we get
h(T) + inf/ <; P(TJ) h(T) + sup/
so P(T,/) = oc iff h(T) = x.
(iv) By the inequality mentioned at the beginning of the proof we have

xe E ( e(S„/> tx)
^SUP
[ xcL e ^ en||/-e||_
This proves (iv).
(v) By Holder’s inequality, if p e [0,1] and E is a finite subset ofX, we have
У eP<S„/)(x) + (t-p)<S„e)(xl < [ у eiS„fMxAP f у <S„«K*>Y * xe £ \xt£ / \«£ J
Therefore P„(T.rf + (1 - p)g.c) < Pn(TJ,c)p ■ Pn(T,g,e)1 ~p and (v) follows.
(vi) is clear from the definition of pressure.
(vii) We have
Pn(T,f + g° T-g,e)
= sup | £ t(s„/)(x)+fl(T"x) - *(x) | £ a |и ^ separated set|
so that
e-2ii„iipn(T, / £) ^ Pn(T) f + g c T _ c) ^ ^H«Hpn(T, /, £).
The result follows from this.
(viii) This follows because P„(T, f + g,e)£ P„{T, /, c) ■ P„(T,g, c).
(ix) II ait.. .,ak are positive numbers with £*= i ot= I then Tj =, a; ^ 1 if с ^ 1, and i a' ^ 1 if с ^ 1. Therefore, if £ is a finite subset of X we have
V -Jc<s"/)<x) <r ( V J^/НхЛ if /■ "> 1
£ t/(S„/)(x) ^ xeE
and
1 herefore, if £. is a hnite si
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