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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 81 82 83 84 85 86 < 87 > 88 89 90 91 92 93 .. 99 >> Next (v T-ŌĆśir) + fsjdvz X KC){-log^C) + a(C)]
'Ō¢ĀiŌĆ£┬░ ' CfŌĆÖvn
1*0
<, log Y eŌĆ£<C) by Lemma 9.9.
CeV ąó"čć 1ŌĆś 0
For each ąĪe\/;=ąŠ T~'r\ choose some xeCso that (SŌĆ×/)(x) = a(C). Since ┬Ż is (n,6) spanning choose y(C) e ┬Ż with d(T'x, Tly{C)) ^ 5, 0 <, j < n ŌĆö 1.
1)9.3 The Variational Principle
219
Then a(C) < (SŌĆ×/)( WC)') + ną│.. Also each ball of radius 6 meets the closures of at most two members of┬╗/ so if čā e E then ]C 6 \/7=o 7"ŌĆÖ,'/|,WC) = >Ō¢Ā} has cardinality at most 2n. Therefore
┬Ż ^to-nt ^ <^2n Y. e<s"/'(>)
ŌĆó ┬╗ ŌĆó ┬╗ ,. - L
Cc V ctVr1!
and so
<- ąŠ i-ąŠ
log I Y eŌĆ£(C>\ ŌĆö ą┐ąĄ< n log 2 + log Y eiSŌĆØ/Hy>
[Cc V T-'4
yeE
Hence
1┬½,(V 7-',) + J/* - i H.( V ą│ą╗) + ; Js./*
< e + log 2 + - log ┬Ż e(SF,/Wv>
n
y-E
^ e + log 2 + - log PŌĆ×(T,/, <5)
n
and therefore
ąøąØ(ąó,┬╗/) + J/ J/; ^ e + log 2 + P(T,f, S)
< e + log2 + P(T,f).
Now hlt(T,┬Ż)<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 SŌĆ×f = Y"^o f┬░ Tl to get ąĖ[ąÉąöąó) + 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 čå e M{X, T) with ą╣ŌĆ×(ąō) + \/<1čå > P(T,f,e), and this clearly implies sup{^(T) + {/e ąøąöąź, T)} ^ P(T,f).
Let EŌĆ× be an (n, e) separated set with
log Y e(S"/,,y);> logąĀą»(ąō,/, e)-1.
Let <rB e M(X) be the atomic measure concentrated on EŌĆ× by the formula
o. =
Y e^fm5y
yeEŌĆ×
┬Ż e(SŌĆ×/)d) xc┬ŻŌĆ×
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 ŌĆö log Pn,(Tj, e) - P(T,f, e) v-*> >h
and {čåą¤]} converges in M(X) to some čå e M(X). By Theorem 6.9 we know čå e M(X, T). We shall show ąøŌĆ×(ąó) + j/ąÉčå > P(T,f, e).
By Lemma 8.5 choose a partition ┬Ż = {At,..., Ak} of (X,J8(X)) so that diam(/4,) < ┬Ż and čå[čüąÉ^ = 0 for 1 < i < k. Since each element of V;=o T contains at most one element of EŌĆ× we have
T~ŌĆś^j + \$SJdan
= X ^({W)((Sn/Kj)-log<Tn({y}))
= log X elSnf)iy) by definition of aŌĆ× and Lemma 9.9:
ye┬ŻŌĆ×
Fix natural members q, n with 1 ┬Ż q < n and as in Remark 2 of ┬¦8.2 define a(j), for 0 < j < q ŌĆö 1, by a(j) = [(n - j)/q], Fix 0 ┬Ż j <, q ŌĆö 1. From Remark 2(i) of ┬¦8.2 we have
ąĖ 1 o(J)-1 fl-1
V t~ŌĆśz = v čé-(ą│┬½+ą╗ v t"'^v V T~lt
i*Q ąō┬╗ 0 f┬╗0 1*S
and S has cardinality at most 2<j. Therefore log ┬Ż ^┬½┬╗ = H0nfv (X/^ŌĆ×
ye┬ŻŌĆ× V/=0 / J
V 7-^+^-fV T-fcA+ fsjda,,
r = 0 \ i = 0 / \keS / J
oO)-l /┬½-1 \ ┬╗
^ I V 7-'┬Ż +2qlogfc + I SŌĆ×/dffn.
r = 0 \i 0 / J
Summing this over j from 0 to q ŌĆö 1 and using Remark 2(iii) of ┬¦8.2 gives 4log X e(S"/K>ŌĆÖ)┬Ż X HcT-J V T~iA + 2q2logk + q {sjdan.
yc┬Żn 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. (*)
ą¤ ąŻ*ąĢŌĆ× \i = 0 / * J
Because n(dAt) = 0 all / we have by Remark 3 of ┬¦8.2 that
Hm ą» (V ąó~ŌĆśąÉ = ąĮ(ą¦\/ T-'A
JŌĆöCO \┬Ż┬½*0 / \J-=0 /
jj9.4 Pressure Determines A/(.V, T)
So replacing n by ns in {*) we get
qP(T,f,e.) ┬Ż + ą® JM<.
Dividing by q and letting q -┬╗ oo gives
P(T,f,e) Z h^T.Q + J/dM < hŌĆ×{T) + J/dAi. ą¤
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) ąĀ(ąó,/) = ąĀ(ąō|ą¤(ąō)./|ą¤,ąó)).
(iii) P(T,f) = ąĀ(ąó(ą┤Ōäó,/|ą┐ Ōäó)-
Proof. The proofs are simple generalisations of the proofs of the corresponding statements in Corollary 8.6. Ō¢Ī
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 ąĪ 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, Ō¢Ā) 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 čå\ąŚą▓(ąź) -* R bp a finite signed measure Then čå e M(X, T) ifflfdv P(T,f) if e CiX, R).
222
9 Topological Pressure and Its Relationship with Invariant Measures
Proof. If čå e M(X, T) then f/dpi < P^T,f) by the variational principle.
Now suppose/ą╗ 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 čü > 0 and n > 0 we have Previous << 1 .. 81 82 83 84 85 86 < 87 > 88 89 90 91 92 93 .. 99 >> Next 