# An introduction to ergodic theory - Walters P.

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(i) Mf(X, T) is convex.

(ii) If h(T) < oo the extreme points of Mf(X, T) are precisely the ergodic members of Mf(X, T).

(iii) If h(T) < oo and Mf(X, T) Ф 0 then Mf(X, T) contains an ergodic measure.

(iv) If the entropy map is upper semi-continuous then Mj{X, T) is compact and non-empty.

(v) // /, £ C{X, R) and if there exists с e R such that f — g — с belongs to the closure of the set {h° T — h\he C(X, R)} in C(X, R), then Mf(X, T) = Mg{X, T).

Proof. The first four parts are proved in the same way as the corresponding parts of Theorem 8.7. To prove (v) we notice that V/z e M(X, T) jf dp = Jg dfx + c. Therefore h^T) + J fdfi = hJ.T) + \g dfi 4- с and P(T,f) = P(T, g) + c. Hence Mf(X, T) = Mg(X, T). □

Part (iv) implies that when T is an expansive homeomorphism every / 6 C(X,R) has an equilibrium state. This also follows from Theorem 9.6 and the proof of the variational principle which also give a way of obtaining an equilibrium state as a limit of atomic measures on separated sets.

The notion of equilibrium state is tied in with the notion of tangent functional to the convex function P(T, ■):C(X,R) -» R.

Definition 9.9. Let T: X -* X be a continuous map of a compact metrisable space with h(T) < oo and let f eC(X,R). A tangent functional to P(T,-) at / is a finite signed measure ц:ЩХ) -► R such that P(T,f + g) — P(T,f) > \g d/л Vg e C(X, R). Let tf(X, T) denote the collection of all tangent functionals to P(T,-) at /.

§9.5 Equilibrium States

225

Remarks

(1) It follows from the Riesz representation theorem (Theorem 6.3) that the dual space C(X, R)* of C(A\ R) can bt identified with the collection of all finite signed measures on (X, jS(X)). This is because each L e С(.Y- R)* is of the form L(f) = V/e C(X,R). (Kingman and Taylor [1] p. 253) Hence we can think of the tangent functionals to P(T, ■) at / as those members L of C(X.R)* satisfying L(q) < P(T,f + g) — P(TJ ) V# 6 C(X, R).

(2) Foreach/e cV , Pjwehaver^A-, T) Ф 0. This follows from Remark

(1) and the Hahn-Banach theorem, since we can extend the identity map on R to an element of C(X,R)* dominated by the convex function g-* P(t.f+g)-P(T,f).

Theorem 9.14, Let T: X -* X be a continuous map of a compact metrisable space X with h(T) < oo and let f e C[X,R). Then M j(X, T) cz tf(X, T) cz ЩХ, T).

Proof. Let y. e Mf{X, T). \tge C(X, R),

P(T, / + g) - P(TJ) > И,(Т) + $fdv+ §gdfi~ K(T) - jfdii = jg J*

by the variational principle. Therefore Mf(X, T) <=. tf(X, T).

We now show tf(X,T)c: M(X,T). Let fietf(X,T). We first show ц takes only non-negative values. Suppose g > 0. If с > 0 we have

J(£ + e)dn = -J-(5 +

> -P(T,f - (g + £)) + P(TJ)

- \P(T,f) - inf((g + £))] + P(TJ) by Theorem 9.7(ii)

= (inf g + e) > 0.

Therefore ]g d/j. :> 0 so ц takes non-negative values. We next show /<(X) = 1. If n e Z then jndp < P(T,f + n) — P(T,f) = n so if n > 1 then ц(Х) ^ 1 and if n ^ — 1 then ц(Х) > 1. Finally we show ц e M(X, T). If n t Z and g e C(X,R\

n J(fif ° T - g)dn ^ P(T,f + n(g ° T - g)) - P(T,f)

= 0 by Theorem 9.13(v).

If n > 0 this gives jg D Tdfi < jg dp and if n < 0 this gives §g ° Tdn> \g dp. Therefore ]g ° T d/л = \g dp so fi e M{X, T). □

With an extra assumption we get the equality of tf(X, T) and Mt(X, T).

Theorem 9.15. Let T:X -* X be a continuous map of a compact metrisable space with h{T) < oo and let f e C(X,R). If the entropy map of T is upper semi-continuous at the members of tj{X,T) then tf(X,T) = Mf(X,T).

226

9 Topological Pressure and Us Relationship wilh Invariant Measures

Proof. It remains to show tf(X, T) c. Mf(X, T). Let pie tf(X,T). Then P(T.f+g)-^(f + g)dti>P(TJ)-lfdn\/geC(X,R) so P{T.h)~\hd,i > P(T,f) — J/dpi V/i e C(A', R). Theorem 9.12 then implies h„(T) > P(T,f) — J/dpi so that P(T,f) = hu(T) + ffdpi by the variational principle. □

Remark. Without the upper semi-continuity assumption one can show tf(X, T) = П„°°= i {M e M(X, T)\hJ,T) + $fd{i>P(TJ)- 1 /«}.

We know that the two-sided shift homeomorphism is expansive and so the following deduction from Theorem 9.15 holds for the shift.

Corollary 9.15.1. Let T:X -> X be a continuous map of a compact metrisable space and suppose the entropy map of T is upper semi-continuous at each point of M(X, T). Then there is a dense subset of C\X,R) such that each member f of this subset has a unique equilibrium state (i.e. M{(X, T) has just one member).

Proof. We use the theorem that a convex function on a separable Banach space has a unique tangent functional at a dense set of points (Dunford and Schwartz [1], p. 450). This combined with Theorem 9.15 gives the result. □

If pi 6 M(X, T) is the only equilibrium state for some / e C(X, R) then this gives a natural way of characterising pi: the only measure with hJ[T) + J/dpt = P(T,f). It turns out in many cases that such a measure pi has very strong ergodic properties. In many cases T is a Bernoulli automorphism of the probability space (A',8ti(X).pi). When T is a specific homeomorphism (such as a shift or an Axiom A difieomorphism) results are known which give conditions on / e C(X, R) to ensure / has a unique equilibrium state (see Bowen [2]). These results are important in the study of difieomorphisms (see§10.1).Forexamplelet T:X -+ X (X = П-°° y- Y = {0,1,... ,k - l})be the shift homeomorphism. Let us denote points of X by x = and

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