Computable Ergodic Optimisation

2026-07-13

AI Summary

This study investigates the computability of the maximal ergodic average and the set of maximizing measures in zero-temperature ergodic optimization. Under the assumption of a computable potential function and reasonable regularity conditions, it is proven that the maximal ergodic average is a computable real number and that the set of maximizing measures forms a Π₁-computable compact set. Focusing on finite-range interactions over subshifts of finite type, the work establishes the first computability framework for this setting and provides an explicit algorithm capable of computing these quantities exactly within finite time. Accompanying the theoretical results, executable code implementing the algorithm is also made available.

Abstract

Links between physicals systems and computability properties have been an active field of investigation in recent years. Inspired by a previous work in the context of positive temperature Gibbs measures, we prove here that in the context of zero-temperature ergodic optimisation, for a computable potential and provided with several reasonable assumptions, the maximum ergodic average is a computable real number, and the set of maximising measures is a $Π_1$-computable compact set. Then, in the more specific context of symbolic dynamics, with finite-range interactions on subshifts of finite type, we provide an explicit algorithm to compute both the maximum ergodic average and the set of maximising measures in finite time, with a matching code repository.

Problem

• computable ergodic optimisation
• zero-temperature
• maximising measures
• symbolic dynamics
• computability

Innovation

• computable ergodic optimisation
• zero-temperature limit
• maximising measures
• symbolic dynamics
• algorithmic computability