Measures of maximal entropy for suspension flows over the full shift

TL;DR

This paper constructs suspension flows over the full shift with prescribed measures of maximal entropy, showing multiple or non-fully supported measures can occur, contrasting with the unique, fully supported measure in the H"older case.

Contribution

It explicitly constructs roof functions for suspension flows that yield specific measures of maximal entropy, including multiple and non-fully supported cases.

Findings

Explicit construction of roof functions with prescribed MME

Multiple measures of maximal entropy can occur in suspension flows

Contrasts with the H"older case where MME is unique and fully supported

Abstract

We consider suspension flows with continuous roof function over the full shift $\Sigma$ on a finite alphabet. For any positive entropy subshift of finite type $Y \subset \Sigma$, we explictly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over $\Sigma$ are exactly the lifts of the measure(s) of maximal entropy for $Y$. In the case when $Y$ is transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If $Y$ has more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a H\"older continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.