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 ${\mu }_f$ 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 Hölder 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 \[\int k dν_{s}=\frac{\sum_{n=1}^{\infty}\sum_{x\in Fix_{n}}e^{sf^{n}(x)-nP(f)}\frac{k^{n}(x)}{n}}{\sum_{n=1}^{\infty}\sum_{x\in Fix_{n}}e^{sf^{n}(x)-nP(f)}},\] where $P(f)$ is the pressure of $f$, $Fi{x}_n$ 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 $cf$ and the corresponding equilibrium state ${\mu }_{cf}$. 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.

References in corpus (1)

• On the zero-temperature limit of Gibbs states