Measures of maximal entropy for suspension flows

TL;DR

This paper investigates the complexity of suspension flows over sub-shifts of finite type, demonstrating the existence and perturbability of flows with uncountably many or unique measures of maximal entropy.

Contribution

It establishes the existence of suspension flows with uncountably many ergodic measures of maximal entropy and shows how to perturb flows to achieve this or a unique measure.

Findings

Existence of flows with uncountably many ergodic measures of maximal entropy

Perturbation methods to control the number of measures of maximal entropy

Flow modifications to achieve unique maximal entropy measure

Abstract

We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any suspension flow defined over a sub-shift of finite type can be perturbed (by an arbitrarily small perturbation) so that the resulting flow has uncountably many ergodic measures of maximal entropy, and that the same can be arranged so that the new flow has a unique measure of maximal entropy.