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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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Proof. We know from Remark 5 that rn(c, A) < s„(e. A') W: > 0.
(i) Let F be a (n,5/2) spanning set for A of cardinality rn(5,‘2.T). Then
and since for each i B(T‘x; <5/2) is a subset of a member of a we have MV’i = o
(ii) If e> 0 and у is an open cover with diam(y) < e then
л — 1
Xs= U П Т-‘ЩГх;5/2)
xeFi=0
T-‘a) <; r„(S/2,X).
174
7 Topological Entropy
(ii) Let £ be a (n,c) separated set of cardinality s„(e. A*). No member of the cover Vf=o T~ly can contain two elements of E so л„1с. X) < ЛМ\/Г=о ■ Т~1у). □
CoroUaty 7.7.1. Let T\X -» X fre a utmtirttmu» чшр «Га смнрмг mctrte space (X, d). Let e > 0. Let ac be the cover of X by all open balls of radius 2c ami let yc be any cover of X by open balls of radius e/2. Then
A'(V < rJbX) < s&M) < n(Vq T-'y^J.
This leads directly to
Theorem 7.8. If T:X —» X is a continuous map of the compact metric space (X,d) then h(T) = h*(T) i.e. the two definitions of topological entropy coincide.
Proof. If f. > 0 and a£,yr are as in Corollary 7.7.1 then h*(T,a,) < f(s, -V, T) < s(e,X,T)< h*(T,yr). If we put f. = I/л and let и-» x the two end terms converge to h*{T) by Theorem 7.6 and the middle terms to /?(T). □
Remark. If we had set up the definition of this section on a uniform space we could have proved Theorem 7.8, for a compact HaudidorfT space.
Corollary 7.7.1 also gives us
Theorem 7.9. If T:X X is a continuous map of a compact metric space (X,d) then
h(T) = lim lim inf- logr„(E, X) = lim lim inf - log *■„(<:, X)
-*0 П-*00 И £ —* 0 Л~* 00 Я
(We know by Corollary 7.5.2 that these formulae hold with “lim inf” replaced by “lim sup”.)
Proof Corollary 7.7.1 gives
h*(T,0Lc) < lim inf - log rn(e, X) ^ lim inf - log .ч„(е, X) < h*(T,y,)
n~* oc 91 n~+ oc 97
and then put e = 1 /к and let к -» oo and use Theorem 7.6. □
We now turn to some more properties of topological entropy.
Theorem 7.10
(i) If (X,d) is a metric space, Te L C(X,d)anclm > 0 then ha( Tm) = m hd(T).
(ii) Let (X,,di), i = 1,2, be a metric space and 7, e UC(Xi.di). Define a metric don X l x .X^by </((*1,x2),(y^yj)) = ^^■{di(xl,yi),d2(x2,y2)\-Then
§7.2 Bowen’s Definition
T, x T2 e UC(Xl x X2. d) and hd{Tt x T2) < + /?dl(T2). 7/ e/rfcer
X, or X2 is compact then hJTt x T2) = /idl(T,) -f hd,(T2).
Proof
(i) Since г„(е, K, Tm) < rmn(£, К, T) we have
- logrn(e. К, ТП < — logr^e, К, T) n mn
and therefore hd(Tm) < m hd(T).
Since T is uniformly continuous, Ve > 0 > 0 such that
d(x,y)<6 implies mux d(TJx, Tjy) < г.
OSjsm-1
So an (n, ^-spanning set for К with respect to Tm is Jso an {mn, e)-spanning set for К with p >ect to T. Hence r„(S, К. T'") > rmn(i, К. T), so mr(£, К, T) < r{5, K, Tm). Therefore
in ■ hd(T, K) < hd( Гт, K).
(ii) Let Ki £ X,- be compact, i = 1. 2. If F, is an (n,*)-spanning set for K, with respect to T- then Ft x F2 is an (H,c)-spanmng set for Kt x K2 with respect to Г, x Г2. Hence
r„(e, x К2, T, x T2) < rn(ElK,, Г,) • r„(t:,K2, T2)
which implies
r(e, Kl x K2, Tl x T2) < r(c, KUTX) + r(v.,K2,T2).
Therefore
kd(Tl x T2, Ki x K2) < hdi(T1,Kl) + hdi(T2,K2).
Let 7t,-:X, x X2 -* X;, i = 1,2 be the projection map. If К £ X, x X2 is compact then Kt = 7i,(K) and K2 = n2iK) are compact and К £ Kt x K2. Hence
hd{T1 x T2,K)< Zi/7'! x T2, R! x K2).
Therefore
MTj x T2)— sup hd(TixT2,K)
CSX, x\-j compact
= sup hd(Tl x T2, A', x KJ
X,SA',
AjEA-j
cpt.
< sup /^,(7',, A.',) 4- sup hdl(T2,h2)
K|SX, k2ex2
cpt. cpt.
176
7 Topological Emropy
Now suppose .Vt is compact. (The prc-of is similar if X, is not compact but X2 is compact.) Since any compact subset of X, x X2 is a subset of X, x K2 for some compact subset K2 of X2, we have hJTl x T,) = supf/z^T, x T2, Xi x K2)\K2 is a compact subset of X2}. Let K2 be a compact subset of X2. Let 5 > 0. If £, is а (и, <5) separated subset of X, and E2 is a (/j, S) separated subset of K2 then El x E2 .s a (n,fijI separated subset of Xx x K2■ Therefore s„(<5, X1 x K2,T1x T2)>sn(6,Xl.Tl)-sn(6,K2,T2) so
s(5,Xy x K2, T, x Tjl^limsup-QogA^X^Tj + logs^K^T;,)]
Л-* fj ^
> liminf-logsn((5, Xu TJ + lim sup - log s„(S, K2, T2).
n~* VC ft n~* 'X ft
Letting 5 -+ 0 we get by Theorem 7.9
MT, x T2,Xl x K2) > + hdi(T2). □
There are examples of homeomorphisms Т,:Х; —> X, (z = 1, 2) of noncompact metric spaces for which x T2) < + ^(Tj). From ihe end of the proof of Theorem 7.10 one can see that one needs
lim sup - [log s„(S, К j, Tj) + log s„(<5, K2,T2)]
n-*co ft
< lim sup - logs„(<5, Klt T,) + lim sup - log.<>„(<5. K2, T2).
n—CC ft n— 00 ft
P. Hulse has shown how to obtain such an example where each X- is the real line equipped with a special metric dt and each Tt is the map x-*x + 1. The idea is that dt differs from the Euclidean metric on some intervals [h, n + 1] and d2 differs from the Euclidean metric on [h, n + 1] for different values of n.
Remark
(16) If T is a homeomorphism with T, T~1 e UC(X, d) then hd(T~*) can differ from hd(T). If T: R -* R is given by Tx = 2x and d is the usual Euclidean metric then we shall see later that hd(T) = log 2 (it is easy to show hd(T) > log 2 by estimating s„(e, f 0,1])). However T~1 decreases distances so hd(T~l) = 0.
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