Zeta measures and Thermodynamic Formalism for temperature zero

Abstract

We address the analysis of the following problem: given a real H\"older potential $f$ defined on the Bernoulli space and $\mu_f$ 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\"older function $f>0$ and a value $s$ such that $0<s<1$, we can associate a shift-invariant probability $\nu_{s}$ such that for each continuous function $k$ we have \[\int k d\nu_{s}=\frac{\sum_{n=1}^{\infty}\sum_{x\in Fix_{n}}e^{sf^{n}(x)-nP(f)}\frac{k^{n}(x)}{n}}{\sum_{n=1}^{\infty}\sum_{x\in Fix_{n}}e^{sf^{n}(x)-nP(f)}},\] where $P(f)$ is the pressure of $f$, $Fix_n$ is the set of solutions of $\sigma^n(x)=x$, for any $n\in \mathbb{N}$, and $f^{n}(x) = f(x) + f(\sigma(x)) + f(\sigma^2(x))+... + f(\sigma^{n-1} (x)).$ We call $\nu_{s}$ a zeta probability for $f$ and $s$. It is known that $\nu_s \to \mu_{f}$, when $s \to 1$. We consider for each value $c$ the potential $c f$ and the corresponding equilibrium state $\mu_{c f}$. What happens with $\nu_{s}$ 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 $c\to \infty, s\to 1$. We will make an assumption: $\lim_{c\to \infty, s\to 1} c(1-s)= L>0$. We do not assume here the maximizing probability for $f$ is unique.