# An introduction to ergodic theory - Walters P.

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Pkoof. Let {Bu..., Внф} be a subcover of p with minimal caroinality. For each /3A-, e a with A{ 2 Bh Therefore {Л,, . , AS(P)] covers X and is a subcover of a. Thus N(a) < N(P). □

(4) H(avP)<H(u) + H(P).

Proof. Let {Ax,... ,/4W(j)} be a subcover of a of minimal cardinality, and {Bt,..., Внф} be a subcover of p of minimal cardinality. Then

(5) If T:X -► X is a continuous map then H(T ‘a) < H(a). If T is also surjective then H(T~la) = Я(а).

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 ‘a of minimal

cardinality then {Ax,... ,/4WT_ia)} also covers A' so N{a) < N(T~la). □

Theorem 7.1. If a is an open cover of X and T:X —> X is continuous then w lim^O/n^V^o Т~‘а) exists.

Proof. Recall that if we set

We h ive

{A; n Bj-Л £ i < N(a), 1 <j<N(p)} is a subcover of a v p, so N(a v p) < N(x)N{P).

□

166

7 Topological Entropy

then by Theorem 4.9 it suffices to show that

a„ + k£a„ + ak fork, n > 1.

We have

(n + k- 1

V T~‘x

f = o

< V T~‘xj + H (т~п V by Remark (4)

<a„ + ak by Remark (5). □

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~‘a

n-tc fl \i=0

Remarks

(6) h(T, a) > 0 by (J).

(7) a < p then h(T, a.) < ЦТ, p).

Proof. If a < /? then \/"=o T~'x < \/;=о T~‘p, so by (3) we have that Я(\/"=о T~\a) < H(V?=o T~% Hence h(T. a) < h(T, /?). □

Note thal if /? s a finite subcover of a then а < p so then ЦТ, a) < ЦТ, P).

(8) h(T,a) < H(a).

Proof. By (4) we have

h(V T-'x) < ”l ЩТ- ‘a)

\i = 0 J i = 0

< n ■ H(a) by (5). □

Definition 7.6. If T:X -*X is continuous, the topological entropy of T is given by:

h(T) = sup /г(Т,а)

a

where a ranges over all open covers of X.

Remarks

(9) ЦТ) >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 = У then /i(T| У) < 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,—*X; are continuous jor i = 1, 2, and if ф:Х1 -» X2 is a continuous map with фХ\ =» A'-, and фТх = Т2ф then h(Tt) > h(T2). If ф is a homeomorphism then h(Tx) = h(T2).

Proof. Let a be an open cover of X2. Then ft(T2,oc) = limitf^V

= \im-н( ф~1"\/ T2'a) by (5) n n \ i=0 J

= lim-#f\/ ф~1Т2,х\

" n \f= о /

= lim - н( V Т^ф-'а)

" м Vi = о J

= h(T„ ф-~' a).

Hence h(T2) < k(Tt). If ф is a homeomorphism then ф~1Тг = Тхф~1 so, by the above, MTj) < /i(T2). □

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). □

168

7 Topnlogical Entropy

§7.2 Bowen’s 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 ■’ 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 d„ on X by d„(x, y) = max0Sl£„_, d(T‘{x), T‘( v)). (The notation does not show the dependence on T.) The open ball centre x and radius r in the metric d„ is pl'=o T_,B(T'x; r).

Definition 7.7. Let л be a natural number, e > 0 and let К be a compact subset of X. A subset F of X is said to (и, e) span К with respect to T if Vx e К 3у e F wi* h d„{x, _y) < e. (i.e.

K <= U "п Т~%Гу,е)).

yeF i= 0

Definition 7.8. If n is a natural number, с > 0 and К is a compact subset of X let r„(c, K) denote the smallest cardinality of any (n, e)-spanning set for К with respect to T. (If we need to emphasise T we shall write ги(к. К, T).)

Remarks

(1) Clearly rjf., K) < oo because the compactness of К implies the covering of К by the open sets P?=o T~'B(T‘x;e), xeX, has a finite subcover.

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