On computability of equilibrium states

Abstract

Equilibrium states are natural dynamical analogs of Gibbs states in thermodynamic formalism. This paper is devoted to the study of their computability in the sense of Computable Analysis. We show that the unique equilibrium state associated to a pair of a computable, topologically exact, distance-expanding, open map $T\colon X\rightarrow X$ and a computable H\"older continuous potential $\varphi\colon X\rightarrow\mathbb{R}$ is always computable. Furthermore, the Hausdorff dimension of the Julia set and the equilibrium state for the geometric potential of a computable hyperbolic rational map are computable. On the other hand, we introduce a mechanism to establish non-uniqueness of equilibrium states. We also present some computable dynamical systems whose equilibrium states are all non-computable.Comment: 41 pages. Reformatted, polished, typos corrected, Theorems D and E reformulated with Section 6 adjusted accordingl