On computability of equilibrium states

Abstract

Equilibrium states are natural dynamical analogues of Gibbs states in thermodynamic formalism. This paper investigates their computability within the framework of Computable Analysis. We show that the unique equilibrium state for a computable, open, topologically exact, distance-expanding map $T\colon X\rightarrow X$ and a computable H\"older continuous potential $\varphi\colon X\rightarrow\mathbb{R}$ is always computable. As an application, we establish the computability of equilibrium states for computable hyperbolic rational maps and their respective geometric potentials. Moreover, we develop a constructive method to exhibit the non-uniqueness of equilibrium states for some dynamical systems. We also present some computable dynamical systems whose equilibrium states are all non-computable.