Computable Ergodic Optimisation
arXiv:2607.11404
The paper shows that for a computable potential in zero‑temperature ergodic optimisation the maximal ergodic average is a computable number and the set of maximizing measures is a Π₁‑computable compact set, and provides an explicit algorithm for symbolic dynamics on subshifts of finite type.
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 -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.