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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 79 80 81 82 83 84 < 85 > 86 87 88 89 90 91 .. 99 >> Next 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 ą║ > 0 and f e C(X,R)
lim sup - logqŌĆ×(T,/, a) = lim sup 1 logqŌĆ×\T,f, \/ T~'a)
nŌĆö r H n-* j H \ ;* ąŠ /
and limŌĆ×^ (l/n) logpŌĆ×(T,f, a) = limŌĆ×_^ (l/n) logpŌĆ×(T,f, \/ŌĆś=o T~la). If T is a homeomorphism and ą║, m > 0 then these formulae hold with \J^o T~ŌĆśa replaced by \/f. _m T~'cc.
and
(k+ lMn'ciŌĆ×+k(T,f, a) ┬Ż qn(r,f, V T-ŌĆśaj Z e<k+ :>u/UqŌĆ×+k(T,f, a)
e~(k+l},l/,,pŌĆ×+k(T,f, a) ┬Ż 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 ^ logpŌĆ×(T,f, T~ '/?) = lim ^ loj┬Ż.pn(T,f,(i)
when T is a homeomorphism and fi is on open cover of A'. Since čā is a finite subcover of V"=o T~'P iff T ŌĆśy is a finite subcover \/"=, T~l/l we have
4AT,f,T-lP)
I inf
ąō. / ┬╗eC
= inf<
Z inf ps"^(v>
( e . ąŻ* ąō >C
V inf
c7y ┬½ff V
( < 7 Jtsf
inf
V is a finite subcovjer of \/ T '({)
|
... l
čā is a finite subcover of \J T \fl\
jccC
čÅ - 1
čā is a finite subcover of \J T~'(l i*0
S: <ąō2||ąø|.
A Similar proof gives
qJLT,f,P)
so the result follows. The result for pŌĆ× follows by a similar proof. The following.result generalises Theorem 7 11.
Ō¢Ī
Theorem 9.6. Let T: X ŌĆö> .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)
nŌĆöan 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 ąó-ąö n- ą░ ąś \ i ŌĆö ŌĆö 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) ŌĆö 2<50). The cover a = {┬Ż(.*,-; (5/2)11 <; i <; /;} has 2d0 for a Lebesgue number so by Theorem 9.2
lim sup - log qŌĆ×(T, f, a) < lim sup - log QŌĆ×( TJ. <5ŌĆ×)
n-* ą░čü 91 n~* x 91
<. lim sup - log PŌĆ×(T,/, <>0) ^ P{T,f).
n ŌĆö* x 91
The result follows by (i) since a is a generator. Ō¢Ī
We can use this to calculate P(T,f) when T is the two-sidtd shift on X = ą¤-┬╗ Y' ąŻ = {Q. L ŌĆö 1}ŌĆÖ and / depends only on the O-th coordinate i.e. /({*ŌĆ×}) = aXo where a0, a,.....at_, e R. If a denotes the natural
generator then qn(T,f,a) = />ŌĆ×(T,/,a) = (e*d + ŌĆó ŌĆó ŌĆó + e┬░k Ō¢Ā)" so P'T,/) = log(t?"0 + ŌĆó ŌĆó ŌĆó + 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, ŌĆó) 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, ŌĆó) is either finite valued or constantly oo.
(iv) |P(T,f,E) - P(T,g,E)| ^ ą”/ - 0||, and so if P(T, Ō¢Ā) < oo, \P(T,f) -P(T,g)\<\\f-g\\.
(v) P(T, -,c) is convex, and so if P(T,-)< oo then P(T. ŌĆó) 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 čü 7> 1 and P\J.tf) 7> cP(T,f) if čü <. 1.
(x) |P(T,/)| ┬Ż P(T,|/|).
\$9.2 Properties ol Pressure
215
Proof. Several nines in the proofs we shall use the simple inequality
sup ą┐.- fa ąö
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
{ąŻ
xe E ( e(SŌĆ×/> tx)
^SUP
[ xcL e ^ en||/-e||_
This proves (iv).
(v) By HolderŌĆÖs inequality, if p e [0,1] and E is a finite subset ofX, we have
ąŻ eP<SŌĆ×/)(x) + (t-p)<SŌĆ×e)(xl < [ čā eiSŌĆ×fMxAP f čā <SŌĆ×┬½K*>Y * xe ┬Ż \xt┬Ż / \┬½┬Ż J
Therefore PŌĆ×(T.rf + (1 - p)g.c) < Pn(TJ,c)p Ō¢Ā 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 | ┬Ż t(sŌĆ×/)(x)+fl(T"x) - *(x) | ┬Ż a |ąĖ ^ separated set|
so that
e-2iiŌĆ×iipn(T, / ┬Ż) ^ Pn(T) f + g c T _ c) ^ ^H┬½Hpn(T, /, ┬Ż).
The result follows from this.
(viii) This follows because PŌĆ×(T, f + g,e)┬Ż PŌĆ×{T, /, c) Ō¢Ā PŌĆ×(T,g, c).
(ix) II ait.. .,ak are positive numbers with ┬Ż*= i ot= I then Tj =, a; ^ 1 if čü ^ 1, and i a' ^ 1 if čü ^ 1. Therefore, if ┬Ż is a finite subset of X we have
V -Jc<s"/)<x) <r ( V J^/ąØčģąø if /Ō¢Ā "> 1
┬Ż t/(SŌĆ×/)(x) ^ xeE
and
1 herefore, if ┬Ż. is a hnite si Previous << 1 .. 79 80 81 82 83 84 < 85 > 86 87 88 89 90 91 .. 99 >> Next 