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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. Let £j > 0. Choose e2 > 0 such that
d'\a, j)<£,=> d(x, y) < f.t and choose e3 > 0 such that
d(x, y) < £3 => d’(x, У) < e2-
Let К be compact. Then
r„{cl,K,d)<rn(t;2,K^d') and r„(c2,K,d') <, rJf.3,K,d).
Hence r(Ei,K,T,d)<r(E2,K,T,d<)<r(E^K,T,d). If Ci-*•(), then £2->0, and £3 -» 0 so we have
hJ(T.K) = hd.(T,K). □
Remarks
(15) The following is an example of two equivalent, but not uniformly equivalent metrics which give different values of entropy for some transformation. Let A' = (0, oo). Define T:(0, со) -» (0. со) by T(.\) = 2л Let d be the Euclidean metric on (0, jo). Then T e UC(X,d) and one can easily show hd(T) > log(2) by estimating the value of r„(£,[ 1,2]). Let d' be the metric which coincides with d on [1,2] but is so that T is ail isometry for d i e. use the fact that the intervals ЯГ ~1, n e Z, partition X and 7’((2"_ 1,2"]) = (2",2’,+ ’]. Then ha (T) = 0 by Remark 11 since T is an isometry for d'. The metrics dd' are equivalent but not uniformly equivalent.
If X is compact and if d and d are equivalent metrics then they are uniformly equivalent. Also each continuous map T: X -* X is uniformly continuous. Therefore if X is a compact metrisable space the entropy of T doe:
172
7 Topological Entropy
not depend on the metric chosen on X (provided that metric induces the topologj of X).
The following will be useful later and it will allow us to simplify the definition of h(T) when X is compact.
Theorem 7.5. Let (X,d) be a metric space and T e UC(X, d). If К а К, и
• • • u Km are all compact subsets of X then h{T; K) < maxt h(T: Kt).
Proof. Certainly s„(e, K) < sn(e., Kt) + ■ • ■ + s„(r„ Km). Fix £ > 0. For each n choose К j(n £) such that s„(c,Ki{r c)) = maxjs^e..КThen s„(c, К) < m • .sn(c,Ki(„.£,) and so,
log .?„(», K) < log m + log ф. Ki(nJ.
Choose n} -* oo such that
— log s (г, К)-lim sup - log s„(a, K)
Hj J r-.ee n
and so that Ki(njt) does not depend on j (i.e.. Kiln.c) - Ki(F)Vj) Theiefore s(£, К, T) < s(e, KiM, T). Choose eq -*■ 0 so that Ki{tq) is constant (= Kio, say). Then h(T: K) < h{T\ Kio) < maxjh(T, Kj). " □
Corollary 7.5.1. Let (X,d) be a metric space and T e UC(X,d). Let a > 0. In order to compute hd{T) is suffices to take the supremum of h(T; K) over those compact sets of diameter less than 5.
Proof. HK is compact it can be covered by a finite number of balls Bu . .. ,Bm of diameter 5/2 and hence h\T\K) < maxlslsm/i(T;K n B,). □
Corollary 7.5.2. If X is a compact metrisable space and d is any metric on X then h(T) = ha(T)=h(T\X).
Proof. If К is a compact subset of X then h(T; K) < h(T; X). It follows from Theorem 7.4 that hd( T) does not depend on d. □
When X is compact we can use Corollary 7.5.2 to simplify the del niiion of h(T). l ake any metric d giving the topology of X. Then
h(T)= lim lim sup - log/-„(с, A") = lim lim sup - log A').
C-»0 ft-» OO И £-* 0 n-*cr> fl
We can give the following interpretation of these expressions. Suppose we want to count the number of orbits of length n (an orbit of length и is a set {*, T(x),..., T"~ '(*)}) but we can only measure to an error e. Then r„(£, X) and s„(e, X) both can be interpreted as the number of orbits of length n up to error £. So as £ —► 0 h(T) is a measurement of the growth rate in n of the number of orbits of length n up to error £.
$7 2 Bowen’s Definition
173
We shall now prove that the definition of h{T) in this section cuincides with that given in i}7.1. when Г is; a continuous map of a compact metrisable space. For the moment let us denote by li*(T) and /i*(T,a) the numbers occurring in the definition of topological entropy using open covers. In a metric sptice (A ,tl) wc define the diamoicr of a cover io be Jiami'a) = sup^a diam(/l), where diamM) denotes the diameter of the set A. If a, у are open covers of X and diam(a) is less than a Lebesgue number for у then у < x The following result is often useful for calculating h*(T).
Theorem 7.6. Let (A,Of) be a compact metric space If [xjf is a sequence of open covers of X with diam(x„) 0 then if h*(T) < oo lim,^,, /i*(T a„) exists and equals h*(T), and if h*(T) = oo then lim,,^ h*(T,oin) = со.
Proof. Suppose h*(T) < со. Let e > 0 be given and choose an open cover у with /i*(7, /) > h*(T) — г.. Let 3 be a Lebesgue number for y. Choose N so that n > N implies diam(a„) < 6. Then у < a„ so h*(T,y) < h*( Г, a„) when n>N. Hence n > TV implies h*(T) > h*(T,oc„) > h*{T) — с so lim,,^*, h*(T.ol„) — h*(T). If h*(T) = со and a > 0 choose an open cover у with h*(T,y) > a and proceed as above to show lim h*(T, x„) = со. □
Corollary 7.6.1. We have h*(T) = Итл_0 (sup/i*(T,a)|diam(a) < <)}.
The next result gives the basic relationship between the two ways of defining topological entropy.
Theorem 7.7. Let T: X —* X be a continuous map of a compact metric space (X,d).
(i) If a is an open cover of X with Lebesgue number S then
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