# An introduction to ergodic theory - Walters P.

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Jn(/ + e)dpt = - J~«(/ + ?)dpi > -P{J,-n(f + c))

^ — [h(T) + sup( — n(f + e))] by Theorem 9.7(ii)

= —h{T) + ninf(/ + e)

> 0 for large n.

Therefore J(/ + e) dpi > 0 so that \f dpi :> 0. Hence pi is a measure.

We now show pt(X) = 1. If n e Z then jndpi £ P(T,n) = h(T) + n, so that pi{X) < 1 + h(T)/n if n > 0 and hence pt(X) < 1, and pi{X) > 1 + h(T)/n if n < 0 and hence pt(X) > 1. Therefore pi(X) = 1.

Lastly we show pi e M(X, T). If n e Z and / e C(X, K), nj(f ° T — f)dpi <, P(T,n(f ° T — /)) = h(T) by Theorem 9.7(vii). If n > 1 then dividing by n and letting n go to oo gives J(/° T — f)dpi < 0, and if и < — 1 then dividing by и and letting n go to — oo gives J(/ ° T — f)dpi> 0. Therefore {/ ° T dpi = \fdpi V/ e C(,Y, R) so pi e M(X, T). □

Theorem 9.11 says that when h(T) < oo the pressure of T detcrnlines the set M(X, T). We now investigate when the pressure of T determines the measure theoretic entropies of T. In the proof we use the fact that if Kx, K2 are disjoint closed convex subsets of a locally convex linear topological space V and if Кг is compact there exists a continuous real-valued linear functional F on V such that F(x) < F(j’) Vx 6 Ku у e K2 (Dunford and Schwartz [1], p. 4)7). In our application V will be C(.Y,R)* x R where C(X,R)* is the dual space of C(X,R) and is equipped with the weak*-topology.

Theorem 9.12. Let T.X -* X be a continuous map of a compact metrisable space with h(T) < oo and let pt0e M(X,T). Then й„0(Г) = inf{P(T,/)— J/dpi0\f e C(X, Я)} iff the emropy map of T is upper semi-continuous at pt0.

Proof. Suppose ftM0(T) = inf{P(T,/) — [fdpi0\fe C(A\i?)}. Let e > 0 be given and choose g e C(X,R) such that P(T,y) — fgdpi0 < h^T) + e/2. Let F„o(g\ e/2) = {pis M(X, T)\\g dpi - jgdpt0\ < e/2}. If pi e Vuo(g\ e/2) then

ЛДТ) < P(T,g) — J*g dpi, by the variational principle,

< P(T,g) - §gdpiо + e/2

< hJT) + e.

Therefore the entropy map is upper semi-continuous at pt0.

Now suppose the entropy map is upper semi-continuous at ц0. By the variational principle we have h^T) ^ inf{P(T,/) — \fdpi0\f e C{X, R)}.

59.5 Equilibrium States

223

We now prove the opposite inequality. Let b > htljT) and let С = [(//./)G \/| A*. T) x К jO < 1 < By Theorem 8.1 С is a convex .et. If we con-

sider С as a subset of С(Л Ri* x R. where the weak "-topology is used on C(X,R)*, then (ц0, )ф С by the upper semi-continuity of the entropy map at Ho- Applying the result quoted above to the disjoint convex sets С and (ц0.) there is a continuous linear functional F.C(X.R)* x R-> R such that F((/i,f)) < F((/t0,b)) V(/<.t) e C. Since we arc using the weak’-topology on C(X.R)* we know that F has the form /-'(/<. t) — }/ <//< + id for some / 6 C(A, R) and some d e R. Therefore | / <//< + th < + tlh V(/i. t) e C, so

j/ilfi + dh„[T) < + dh\/fi 6 Л/| A. T). If we put /i = /i„ then <//!„„( T) <

db so d > 0. Hence

K(T) + J L dn < b + J ^ dv0 Чц £ MIX, T)

so, Dy the variational principle,

P(T,f/d) £b + jf dn0.

Rearranging gives

b > P(T,f/d) - jf/dd,t0 > inf|p(T,</) -Therefore h^T) ^ inf{P(T.</) - \g £ C(X. R)\. □

Remarks

(1) The same proof shows that if T: X -» A is a continuous map with h{T) < cc and /i0 £ M(X, T) then inf\P(T,f) — 6 C(. , R)',=

sup[limsup„_ , h,JT)\1/<„! is a sequence in Aft A'. T) with ц„ -»//„ J

(2i Theorems 9.11 and 9.12 show that if the entropy map of T is upper semi-continuous at every point of Л/(А, T) and /i(T) < x then P(T, ) determines the set M( A, T) and the entropy hlt(T) for all ц £ A/( A, T). Combining this with the variational principle wc sec that when the entropy map of T is upper semi-continuous on M(X, T) (for example, when T is an expansive homeomorphism (Theorem 8.2.)) and h(T) < oo then knowledge of P(T, ):C(A, R)-> R is equivalent to the knowledge of M(X, T) and hr(T) for all // e M(X, T). Hence the pressure contains a lot of information.

(3) It is not difficult to show that if /i(7) < oo then T is uniquely ergodic iff P(T, ):C(A,R)-» R is (Frcchet) diflereni.able at each point of C(X,R).

§9.5 Equilibrium States

The variational principle gives a natural way of selecting members of M(X, T). The concept extends the idea of measure with maximal entropy.

224

9 Topological Pressure and Its Relationship with Invanant Measures

Definition 9.8. Let T:X -* X be a continuous map of a compact metrisable space X and let / e C(X, R). A member ц of M(.Y, T) .s called an equilibrium state for f if P(T,f) = h^T) + [f dy.. Let Mf{X, T) denote the collection of all equilibrium states for /.

Remarks

(1) A measure with maximal entropy is precisely an equilibrium state for 0. Hence jvfmax(X, T) is the same as M0(A', T).

(2) As we know from §8.3 the set Mf(X, T) can be empty.

(3) If Л(7") = oo then Mf(X, T) = {/< e ЩХ, T)\hJLT) = oo} V/e C(X,R), so X, T) Ф 0 by Theorem 8.7(iv).

We have the following generalisation of Theorem 8.7.

Theorem 9.13. Let T:X —> X be a continuous map of a compact metrisable space and let f e C(X,R). Then

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