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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 62 63 64 65 66 67 < 68 > 69 70 71 72 73 74 .. 99 >> Next Pkoof. Let {Bu..., ðÆð¢Ðä} be a subcover of p with minimal caroinality. For each /3A-, e a with A{ 2 Bh Therefore {ðø,, . , AS(P)] covers X and is a subcover of a. Thus N(a) < N(P). Ôûí
(4) H(avP)<H(u) + H(P).
Proof. Let {Ax,... ,/4W(j)} be a subcover of a of minimal cardinality, and {Bt,..., ðÆð¢Ðä} be a subcover of p of minimal cardinality. Then
(5) If T:X -Ôû║ X is a continuous map then H(T ÔÇÿa) < H(a). If T is also surjective then H(T~la) = ð»(ð░).
Proof. If {At,..., AN(a)} is a subcover of a of minimal cardinality then {T~lAu ..., T~lANM} is a subcover of T-Ia, so N(T~1a) < N(ot). If T is surjective and {T is a subcover of T ÔÇÿa of minimal
cardinality then {Ax,... ,/4WT_ia)} also covers A' so N{a) < N(T~la). Ôûí
Theorem 7.1. If a is an open cover of X and T:X ÔÇö> X is continuous then w lim^O/n^V^o ðó~ÔÇÿð░) exists.
Proof. Recall that if we set
We h ive
{A; n Bj-ðø ┬ú i < N(a), 1 <j<N(p)} is a subcover of a v p, so N(a v p) < N(x)N{P).
Ôûí
166
7 Topological Entropy
then by Theorem 4.9 it suffices to show that
aÔÇ× + k┬úaÔÇ× + ak fork, n > 1.
We have
(n + k- 1
V T~ÔÇÿx
f = o
< V T~ÔÇÿxj + H (Ðé~ð┐ V by Remark (4)
<aÔÇ× + ak by Remark (5). Ôûí
Definition 7.5. If a is an open cover of X and T:X -* X is a continuous map then the entropj of T relative to n is given by:
1 (n~1 h(T, a) = lim - Hi \f T~ÔÇÿa
n-tc fl \i=0
Remarks
(6) h(T, a) > 0 by (J).
(7) a < p then h(T, a.) < ðªðó, p).
Proof. If a < /? then \/"=o T~'x < \/;=ð¥ T~ÔÇÿp, so by (3) we have that ð»(\/"=ð¥ T~\a) < H(V?=o T~% Hence h(T. a) < h(T, /?). Ôûí
Note thal if /? s a finite subcover of a then ð░ < p so then ðªðó, a) < ðªðó, P).
(8) h(T,a) < H(a).
Proof. By (4) we have
h(V T-'x) < ÔÇØl ð®ðó- ÔÇÿa)
\i = 0 J i = 0
< n Ôûá H(a) by (5). Ôûí
Definition 7.6. If T:X -*X is continuous, the topological entropy of T is given by:
h(T) = sup /ð│(ðó,ð░)
a
where a ranges over all open covers of X.
Remarks
(9) ðªðó) >0.
(10) In the definition of h(T) one can take the supremum over finite open covers of X. This follows from (7).
\$7.1 Definition Using Open Com rs
167
(11) h(l) = 0 where I is the identity map of X.
(12) If Y is a dosed subset of X and TY = ðú then /i(T| ðú) < h(T).
The next result shows that topological entropy is an invariant of topological conjugacy.
Theorem 7.2. If Xj, X2 aie compact spacts and T^.X,ÔÇö*X; are continuous jor i = 1, 2, and if Ðä:ðÑ1 -┬╗ X2 is a continuous map with ÐäðÑ\ =┬╗ A'-, and ÐäðóÐà = ðó2Ðä then h(Tt) > h(T2). If Ðä is a homeomorphism then h(Tx) = h(T2).
Proof. Let a be an open cover of X2. Then ft(T2,oc) = limitf^V
= \im-ð¢( Ðä~1"\/ T2'a) by (5) n n \ i=0 J
= lim-#f\/ Ðä~1ðó2,Ðà\
" n \f= ð¥ /
= lim - ð¢( V ðó^Ðä-'ð░)
" ð╝ Vi = ð¥ J
= h(TÔÇ× Ðä-~' a).
Hence h(T2) < k(Tt). If Ðä is a homeomorphism then Ðä~1ðóð│ = ðóÐàÐä~1 so, by the above, MTj) < /i(T2). Ôûí
In the next section we shall give a definition of lt{T) that does not require X to be compact and we shall prove properties of h(T) in this more general setting. However, one result that is false when X is not compact is the following.
Theorem 7.3. If T: X -* X is a homeomorphism of a compact space X then h(T) = h(T-i).
Proof
h(T,a) = lim - H^\f T~
= lim ^ ^T"'1 ^ \/ T"ia)j by Remark (5)
= h(T-\a). Ôûí
168
7 Topnlogical Entropy
┬º7.2 BowenÔÇÖs Definition
In this section we give the definition of topological entropy using separating and spanning sets This was done by Dinaburg ar.d by Bowen but Howj;n also gave the definition v hen the space X is not compact and this will prove useful later. We shall give the definition when X is a metric sp-ice but the definition can easily be formulated when X is a uniform space.
In this section (X,d) is ÔûáÔÇÖ metric space, not necessarily compact. The open ball centre v radius r u .. be denoted by B(.x;r), and the closed ball by B(x; r). We shall define topological entropy for uniformly continuous maps T:X -┬╗ X. The space of all uniformly continuous maps of the metric space (X, d) will be denoted by UC{X, d). Our definitions will depend on the metric d on X; we shall see later what the dependence on d is.
Throughout this section T will denote a fixed member of UC(X,d). If n is a natural number we can define a new metric dÔÇ× on X by dÔÇ×(x, y) = max0Sl┬úÔÇ×_, d(TÔÇÿ{x), TÔÇÿ( v)). (The notation does not show the dependence on T.) The open ball centre x and radius r in the metric dÔÇ× is pl'=o T_,B(T'x; r).
Definition 7.7. Let ð╗ be a natural number, e > 0 and let ðÜ be a compact subset of X. A subset F of X is said to (ð©, e) span ðÜ with respect to T if Vx e ðÜ 3Ðâ e F wi* h dÔÇ×{x, _y) < e. (i.e.
K <= U "ð┐ ðó~%ðôÐâ,ðÁ)).
yeF i= 0
Definition 7.8. If n is a natural number, Ðü > 0 and ðÜ is a compact subset of X let rÔÇ×(c, K) denote the smallest cardinality of any (n, e)-spanning set for ðÜ with respect to T. (If we need to emphasise T we shall write ð│ð©(ð║. ðÜ, T).)
Remarks
(1) Clearly rjf., K) < oo because the compactness of ðÜ implies the covering of ðÜ by the open sets P?=o T~'B(TÔÇÿx;e), xeX, has a finite subcover. Previous << 1 .. 62 63 64 65 66 67 < 68 > 69 70 71 72 73 74 .. 99 >> Next 