Books in black and white
 Books Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics

An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 82 83 84 85 86 87 < 88 > 89 90 91 92 93 94 .. 99 >> Next Jn(/ + e)dpt = - J~┬½(/ + ?)dpi > -P{J,-n(f + c))
^ ŌĆö [h(T) + sup( ŌĆö n(f + e))] by Theorem 9.7(ii)
= ŌĆöh{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 ┬Ż 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 ŌĆö f)dpi <, P(T,n(f ┬░ T ŌĆö /)) = h(T) by Theorem 9.7(vii). If n > 1 then dividing by n and letting n go to oo gives J(/┬░ T ŌĆö f)dpi < 0, and if ąĖ < ŌĆö 1 then dividing by ąĖ and letting n go to ŌĆö oo gives J(/ ┬░ T ŌĆö f)dpi> 0. Therefore {/ ┬░ T dpi = \fdpi V/ e C(,Y, R) so pi e M(X, T). Ō¢Ī
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 ąÜą│ is compact there exists a continuous real-valued linear functional F on V such that F(x) < F(jŌĆÖ) Vx 6 Ku čā e K2 (Dunford and Schwartz , 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 ą╣ŌĆ×0(ąō) = inf{P(T,/)ŌĆö J/dpi0\f e C(X, ą»)} iff the emropy map of T is upper semi-continuous at pt0.
Proof. Suppose ftM0(T) = inf{P(T,/) ŌĆö [fdpi0\fe C(A\i?)}. Let e > 0 be given and choose g e C(X,R) such that P(T,y) ŌĆö fgdpi0 < h^T) + e/2. Let FŌĆ×o(g\ e/2) = {pis M(X, T)\\g dpi - jgdpt0\ < e/2}. If pi e Vuo(g\ e/2) then
ąøąöąó) < P(T,g) ŌĆö J*g dpi, by the variational principle,
< P(T,g) - ┬¦gdpiąŠ + e/2
< hJT) + e.
Therefore the entropy map is upper semi-continuous at pt0.
Now suppose the entropy map is upper semi-continuous at čå0. By the variational principle we have h^T) ^ inf{P(T,/) ŌĆö \fdpi0\f e C{X, R)}.
59.5 Equilibrium States
223
We now prove the opposite inequality. Let b > htljT) and let ąĪ = [(//./)G \/| A*. T) x ąÜ jO < 1 < By Theorem 8.1 ąĪ is a convex .et. If we con-
sider ąĪ as a subset of ąĪ(ąø Ri* x R. where the weak "-topology is used on C(X,R)*, then (čå0, ą)čä ąĪ by the upper semi-continuity of the entropy map at Ho- Applying the result quoted above to the disjoint convex sets ąĪ and (čå0.ą) 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 weakŌĆÖ-topology on C(X.R)* we know that F has the form /-'(/<. t) ŌĆö }/ <//< + id for some / 6 C(A, R) and some d e R. Therefore | / <//< + th < + tlh V(/i. t) e C, so
j/ilfi + dhŌĆ×[T) < + dh\/fi 6 ąø/| A. T). If we put /i = /iŌĆ× then <//!ŌĆ×ŌĆ×( T) <
db so d > 0. Hence
K(T) + J L dn < b + J ^ dv0 ą¦čå ┬Ż MIX, T)
so, Dy the variational principle,
P(T,f/d) ┬Żb + jf dn0.
Rearranging gives
b > P(T,f/d) - jf/dd,t0 > inf|p(T,</) -Therefore h^T) ^ inf{P(T.</) - \g ┬Ż C(X. R)\. Ō¢Ī
Remarks
(1) The same proof shows that if T: X -┬╗ A is a continuous map with h{T) < cc and /i0 ┬Ż M(X, T) then inf\P(T,f) ŌĆö 6 C(. , R)',=
sup[limsupŌĆ×_ , h,JT)\1/<ŌĆ×! is a sequence in Aft A'. T) with čåŌĆ× -┬╗//ŌĆ× J
(2i Theorems 9.11 and 9.12 show that if the entropy map of T is upper semi-continuous at every point of ąø/(ąÉ, T) and /i(T) < x then P(T, ) determines the set M( A, T) and the entropy hlt(T) for all čå ┬Ż 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 čå 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 ąø(7") = oo then Mf(X, T) = {/< e ą®ąź, T)\hJLT) = oo} V/e C(X,R), so X, T) ąż 0 by Theorem 8.7(iv).
We have the following generalisation of Theorem 8.7.
Theorem 9.13. Let T:X ŌĆö> X be a continuous map of a compact metrisable space and let f e C(X,R). Then Previous << 1 .. 82 83 84 85 86 87 < 88 > 89 90 91 92 93 94 .. 99 >> Next 