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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 64 65 66 67 68 69 < 70 > 71 72 73 74 75 76 .. 99 >> Next Proof. Let ┬Żj > 0. Choose e2 > 0 such that
d'\a, j)<┬Ż,=> d(x, y) < f.t and choose e3 > 0 such that
d(x, y) < ┬Ż3 => dŌĆÖ(x, ąŻ) < e2-
Let ąÜ be compact. Then
rŌĆ×{cl,K,d)<rn(t;2,K^d') and rŌĆ×(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-*ŌĆó(), then ┬Ż2->0, and ┬Ż3 -┬╗ 0 so we have
hJ(T.K) = hd.(T,K). Ō¢Ī
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, čüąŠ) -┬╗ (0. čüąŠ) by T(.\) = 2ą╗ 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 rŌĆ×(┬Ż,[ 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 ą»ąō ~1, n e Z, partition X and 7ŌĆÖ((2"_ 1,2"]) = (2",2ŌĆÖ,+ ŌĆÖ]. 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:
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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 ąÜ ą░ ąÜ, ąĖ
ŌĆó ŌĆó ŌĆó u Km are all compact subsets of X then h{T; K) < maxt h(T: Kt).
Proof. Certainly sŌĆ×(e, K) < sn(e., Kt) + Ō¢Ā ŌĆó Ō¢Ā + sŌĆ×(rŌĆ× Km). Fix ┬Ż > 0. For each n choose ąÜ j(n ┬Ż) such that sŌĆ×(c,Ki{r c)) = maxjs^e..ąÜThen sŌĆ×(c, ąÜ) < m ŌĆó .sn(c,Ki(ŌĆ×.┬Ż,) and so,
log .?ŌĆ×(┬╗, K) < log m + log čä. Ki(nJ.
Choose n} -* oo such that
ŌĆö log s (ą│, ąÜ)-lim sup - log sŌĆ×(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(┬Ż, ąÜ, T) < s(e, KiM, T). Choose eq -*Ō¢Ā 0 so that Ki{tq) is constant (= Kio, say). Then h(T: K) < h{T\ Kio) < maxjh(T, Kj). " Ō¢Ī
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,). Ō¢Ī
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 ąÜ 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. Ō¢Ī
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/-ŌĆ×(čü, A") = lim lim sup - log A').
C-┬╗0 ft-┬╗ OO ąś ┬Ż-* 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 ąĖ is a set {*, T(x),..., T"~ '(*)}) but we can only measure to an error e. Then rŌĆ×(┬Ż, X) and sŌĆ×(e, X) both can be interpreted as the number of orbits of length n up to error ┬Ż. So as ┬Ż ŌĆöŌ¢║ 0 h(T) is a measurement of the growth rate in n of the number of orbits of length n up to error ┬Ż.
\$7 2 BowenŌĆÖs Definition
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We shall now prove that the definition of h{T) in this section cuincides with that given in i}7.1. when ąō 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, čā are open covers of X and diam(a) is less than a Lebesgue number for čā then čā < 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(xŌĆ×) 0 then if h*(T) < oo lim,^,, /i*(T aŌĆ×) exists and equals h*(T), and if h*(T) = oo then lim,,^ h*(T,oin) = čüąŠ.
Proof. Suppose h*(T) < čüąŠ. Let e > 0 be given and choose an open cover čā with /i*(7, /) > h*(T) ŌĆö ą│.. Let 3 be a Lebesgue number for y. Choose N so that n > N implies diam(aŌĆ×) < 6. Then čā < aŌĆ× so h*(T,y) < h*( ąō, aŌĆ×) when n>N. Hence n > TV implies h*(T) > h*(T,ocŌĆ×) > h*{T) ŌĆö čü so lim,,^*, h*(T.olŌĆ×) ŌĆö h*(T). If h*(T) = čüąŠ and a > 0 choose an open cover čā with h*(T,y) > a and proceed as above to show lim h*(T, xŌĆ×) = čüąŠ. Ō¢Ī
Corollary 7.6.1. We have h*(T) = ąśčéą╗_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 ŌĆö* 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 Previous << 1 .. 64 65 66 67 68 69 < 70 > 71 72 73 74 75 76 .. 99 >> Next 