Zeta measures and Thermodynamic Formalism for temperature zero

Abstract

We address the analysis of the following problem: given a real Hölder 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 Hölder 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

$$ \int {kd} v_s = \frac{{\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{x \in Fix_n } {e^{sf^n (x) - nP(f)\frac{{k^n (x)}} {n}} } } }} {{\sum\nolimits_{n = 1}^\infty {\sum\nolimits_{x \in Fix_n } {e^{sf^n (x) - nP(f)} } } }}, $$

, where P(f) is the pressure of f, Fix _n is the set of solutions of σ ⁿ(x) = x, for any n ∈ 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.