On the speed of convergence of the pressure function at zero temperature

TL;DR

This paper establishes a lower bound on the rate at which the pressure function approaches the limit entropy at zero temperature, linking it to Peierls barriers and extending previous results to more general settings.

Contribution

It characterizes the exponential convergence rate of the pressure function at zero temperature using Peierls barriers, generalizing prior specific cases.

Findings

Pressure function cannot converge faster than exponential rate.

The convergence rate is characterized by Peierls barriers.

Extends previous results to broader classes of systems.

Abstract

We prove here that the pressure function cannot converge to the limit entropy at zero temperature faster than some exponential rate. Furthermore, we characterize this limit rate via an expression involving the Peierls barriers between the irreducible components of the Aubry set. This extends and completes results from [8] and [7]. In the first one, an exact exponential speed of convergence was proved, under the assumption that the Aubry set is a subshift of finite type. In the later one, a rate was given but without interpretation in term of Thermodynamical quantities of the system.