Computable thermodynamic formalism

TL;DR

This paper explores the computability of equilibrium states in thermodynamic formalism within dynamical systems, providing methods to verify computability and applying them to specific classes like rational maps.

Contribution

It introduces two new approaches for establishing the computability of equilibrium states in nonuniformly expanding systems, expanding the theoretical framework.

Findings

Established computability of equilibrium states for Misiurewicz-Thurston rational maps.

Developed methods applicable to systems with non-upper semicontinuous entropy functions.

Provided a theoretical foundation for computability in thermodynamic formalism.

Abstract

We investigate the theory of thermodynamic formalism from the perspective of computable analysis, with a special focus on the computability of equilibrium states. Specifically, we develop two complementary general approaches to verify the computability of equilibrium states for nonuniformly expanding computable dynamical systems. The first approach applies to dynamical systems whose topological pressure functions admit effective approximations and whose measure-theoretic entropy functions are upper semicontinuous. As a concrete application, we establish the computability of the equilibrium states for Misiurewicz-Thurston rational maps with H\"older continuous potentials. The second approach exploits prescribed Jacobians of equilibrium states through a local analysis and applies to settings where the measure-theoretic entropy functions may lack upper semicontinuity.