# An introduction to ergodic theory - Walters P.

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§8.6 Definition of Measure-Theoretic Entropy Using the Metrics d„

Let (X,d) be a compact metric space and let T .X -* X be a homeomorphism. In §7.2 we introduced the metrics dn on X by d„(x, y) = max0sis„., d{T‘(x), T‘(y)). We then defined r„(c,X) to be the minimum number of r.-balls, in the d„ metric, whose union covers X, and we showed h(T) = lim£^0

206

8 Relationship Between Topological Entropy and Measure-Theoretic Entropy

limsupB_,. (1 In) log rjt.. X) = lim,^0 liming.., (1 n)k>grje, X). Л. P Katok has given an analogous description of measurc-theoretic entropy.

Theorem 8.19. Let T:X —► X be a Homeomorphism of the compact metric spacj (X, cl). Let m e M(X, T) and Itt m be ergodic. For i: > 0, <5 > 0 let r„(e,S,m) denote the minimum number of E-balls in the dn metric whose union has m-measure more than or equal to 1—5. Thun, for each 5 > 0, we have

hm(T) = lim lim sup - Vog in(r.,6, m) — lim lim inf- log r„(c. 5,m).

e -• 0 n— x П t“*0 n— x И

We refer to Katok [1] for the proof.

CHAPTER 9

Topological Pressure and Its Relationship with Invariant Measures

Let T\X -* X be a continuous transformation of a compact metric space (X,d). Let C(X, R) denote the Banach algebra of real-valued continuous functions of X equipped with the supremum norm. The topological pressure of T will be a map PIT, ):C(X,R}~* R u {oo} which will have good properties relative to the structures on C(\.R). It contains topological entropy in the sense that P(T,0) = h(T) where 0 denotes the member of C’(X,R) which is identically zero. A generalisation of the variational principle of §8.2 is true and this sometimes gives a natural way of choosing important members of M(X, T). In this theory ideas from mathematical statistical mechanics are used and the theory has important applications to other fields. We shall mention in §10.1, one important application to differentiable dynamical systems.

The concept of pressure in this type of setting was introduced by Ruclle[l] and studied in the general case in Walters [3].

§9.1 Topological Pressure

Let (X,d) be a compact metric space C(X, R) the space of real-valued continuous functions of X and T: X -* X a continuous transformation. We shall use natural logarithms. The definition of pressure can be given by using open covers or spanning sets or separated *ets. Since X is compact we can generalise the definition of h(T) given in the remarks following Corollary 7.5.2. For / e C(X, R) and n^lwe denote £7=o /(^x) by (S„/ )(x).

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9 Topological Pressure and Its. Relationship with Invariant Measures

Definition 9.1. For / e C(X, R), n ^ 1 and с > 0 put

Qn(T,f,£) — inf< Y. e,SnfM| F is a (n,r.) spanning set for X

[xeF

Remarks

(1) 0 < Qn(T,f,E) < Це^ЦгДс, X) < x (see Remark (1) of §7.2).

(2) If e, < s2 then Q„[T,f,Et) > Q„{T,f.E2).

(3) Qn(7,0, £) = rn(£, X).

(4) In Definition 9.1 it suffices to take the mfinium over those (n,r.) spanning sets which don’t have proper subsets that (n,c) span X. This is btcause

e(S„/Hx) > Q

Definition 9.2. For / e C(X, R) and с > 0 put

Q(T,/,e) = lim sup - logQJT,/,e).

Я “■* X ft

Remarks

(5) Q(T,f,e) ^ ||/|| + r(c. X, T)< oc (see Remark (1) and Theorem 7.7(ii)X

(6) IfCi < e2 then Q{T,f,£i) ^ Q(7\/,E2)(by Remark (2)).

Definition 9.3. If / e C(X,R) let P(T,f) denote lim£^0Q(T,/,c).

Remarks

(7) Bv Remark (6), P(T,f) exists but could be oo.

Definition 9.4. The map P(7, ):C(X,R)-*i!u {oo} defined above is called the topological pressure of T.

Clearly P(T,0) = h(T). We shall obtain some equivalent ways of giving the definition.

Definition 9.5. For / e C(X, R), n ^ 1 and e > 0 put

PJiT,f,£) = sup< Z e(S"/)(x)|E is a (n,c) separated subset of X U e£

Remarks

(8) If c, < e2 then Pn(T,/,£,) ^ P„(T,/,e2).

(9) Р„( T, 0, e) = s„(e, X).

(10) In Definition 9.5 it suffices to take the supremum over all the (и,е) separated sets which fail to be (n,s) separated when any point of X is added. This is because e(Snf]U) > 0.

(11) We have Q„(T,f,E) ^ P„(T,/,e). This follows from Remark (10) and the fact that а (и, e) separated set which cannot be enlarged to a (n, e) separated set must be a (n, e) spanning set for X.

59.1 Topol Jgical Pressure

209

(12) If й > 0 is such that d(x. у) < г./2 implies |/(v) — /(\ )|<<'> 'ben PJT.f.e) <; e^QjTJ.j 2).

Proof. Let E be а (я,*) separated set and F a \>ur. 2) spanning set. Define ф:E —► F by choosing, for each x e £, some point ф( v) e F with d„(x, ф(х)) <, e,2 (using the notation dn(x,y) = max0:SJSll_, t/(T‘t\-), T‘( v))- Then </> is injective so

£ (jWKj) ^ ^ j mjn j £

yeF уефК \«£ / ag£

(13) Q{T,f,c)<, P{TJ,c) (by Remark (11)).

(14) If «5 is such that J(x,y) < e/2 implies |/(x) — /(y)| < f> then P(T,/,e) ^ <5 + (2(T,/,£) (by Remark (12)).

(15) If e, < £2 then P(T,/,£,)> P(T4/,£2).

Theorem 9.1. If f e C(X,R) then P(T,f) = limt^0 P(T,f,e).

Proof. The limit exists by Remark 15. By Remark 13 we have P(T,f) <, lim£_0 P\T,f,s).

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