Existence of the zero temperature limit of equilibrium states on topologically transitive countable Markov shifts

TL;DR

This paper proves the existence of the zero temperature limit of equilibrium states for certain countable Markov shifts and potentials, extending understanding of thermodynamic limits in dynamical systems.

Contribution

It establishes the existence of the zero temperature limit for equilibrium states on countable Markov shifts under specific conditions, including summability and finite Gurevich pressure.

Findings

Limit of equilibrium states exists as temperature approaches zero.

Examples show zero temperature limits exist under broader conditions.

Provides new insights into thermodynamic limits in symbolic dynamics.

Abstract

Consider a topologically transitive countable Markov shift $\Sigma$ and a summable Markov potential $\phi$ with finite Gurevich pressure and $\mathrm{Var}_1(\phi) < \infty$. We prove existence of the limit $\lim_{t \to \infty} \mu_t$ in the weak$^\star$ topology, where $\mu_t$ is the unique equilibrium state associated to the potential $t\phi$. Besides that, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.