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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 83 84 85 86 87 88 < 89 > 90 91 92 93 94 95 .. 99 >> Next (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) –§ 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 —Ā e R such that f ‚ÄĒ g ‚ÄĒ —Ā belongs to the closure of the set {h¬į T ‚ÄĒ 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- —Ā and P(T,f) = P(T, g) + c. Hence Mf(X, T) = Mg(X, T). ‚Ė°
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, ‚Ė†):C(X,R) -¬Ľ 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 —Ü:–©–•) -‚Ėļ R such that P(T,f + g) ‚ÄĒ P(T,f) > \g d/–Ľ Vg e C(X, R). Let tf(X, T) denote the collection of all tangent functionals to P(T,-) at /.
¬ß9.5 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 –°(.Y- R)* is of the form L(f) = V/e C(X,R). (Kingman and Taylor  p. 253) Hence we can think of the tangent functionals to P(T, ‚Ė†) at / as those members L of C(X.R)* satisfying L(q) < P(T,f + g) ‚ÄĒ P(TJ ) V# 6 C(X, R).
(2) Foreach/e cV , Pjwehaver^A-, T) –§ 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 –©–•, T).
Proof. Let y. e Mf{X, T). \tge C(X, R),
P(T, / + g) - P(TJ) > –ė,(–Ę) + \$fdv+ ¬ßgdfi~ 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 —Ü takes only non-negative values. Suppose g > 0. If —Ā > 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 —Ü takes non-negative values. We next show /<(X) = 1. If n e Z then jndp < P(T,f + n) ‚ÄĒ P(T,f) = n so if n > 1 then —Ü(–•) ^ 1 and if n ^ ‚ÄĒ 1 then —Ü(–•) > 1. Finally we show —Ü e M(X, T). If n t Z and g e C(X,R\
n J(fif ¬į T - g)dn ^ P(T,f + n(g ¬į 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 ¬ßg ¬į Tdn> \g dp. Therefore ]g ¬į T d/–Ľ = \g dp so fi e M{X, T). ‚Ė°
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) ‚ÄĒ J/dpi V/i e C(A', R). Theorem 9.12 then implies h‚Äě(T) > P(T,f) ‚ÄĒ J/dpi so that P(T,f) = hu(T) + ffdpi by the variational principle. ‚Ė°
Remark. Without the upper semi-continuity assumption one can show tf(X, T) = –ü‚Äě¬į¬į= i {M e M(X, T)\hJ,T) + \$fd{i>P(TJ)- 1 /¬ę}.
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 , p. 450). This combined with Theorem 9.15 gives the result. ‚Ė°
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 ). These results are important in the study of difieomorphisms (see¬ß10.1).Forexamplelet T:X -+ X (X = –ü-¬į¬į y- Y = {0,1,... ,k - l})be the shift homeomorphism. Let us denote points of X by x = and Previous << 1 .. 83 84 85 86 87 88 < 89 > 90 91 92 93 94 95 .. 99 >> Next 