Zeta measures and Thermodynamic Formalism for temperature zero

摘要

We address the analysis of the following problem: given a real H?lder potential f defined on the Bernoulli space and μ_f its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a H?lder function f > 0 and a value s such that 0 < s < 1, we can associate a shift-invariant probability ν_s such that for each continuous function k we have, where P(f) is the pressure of f, Fix_n is the set of solutions of σ~n(x) = x, for any n ∈ ?, and f~n(x) = f(x) + f(σ (x)) + . . . + f(σ~(n-1)(x)). We call νs a zeta probability for f and s, because it can be obtained in a natural way from the dynamical zeta-functions. From the work of W. Parry and M. Pollicott it is known that ν_s → μ_f, when s → 1. We consider for each value c the potential c f and the corresponding equilibrium state μ_(cf). What happens with ν_s when c goes to infinity and s goes to one? This question is related to the problem of how to approximate the maximizing probability for f by probabilities on periodic orbits. We study this question and also present here the deviation function I and Large Deviation Principle for this limit c → ∞, s → 1. We will make an assumption: for some fixed L we have lim_(c→∞, s→1)c(1 - s) = L > 0. We do not assume here the maximizing probability for f is unique in order to get the L. D. P.