Upper semi-continuity of entropy in non-compact settings

TL;DR

This paper proves the upper semi-continuity of entropy for countable Markov shifts with finite entropy in non-compact spaces, with applications to dynamical systems like positive entropy diffeomorphisms.

Contribution

It establishes upper semi-continuity of the entropy map in non-compact settings and explores implications for measures of maximal entropy.

Findings

Entropy map is upper semi-continuous at ergodic measures

Applications to systems coded by countable Markov shifts

Discussion on existence of measures of maximal entropy

Abstract

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive entropy diffeomorphisms on compact manifolds, are given. We also discuss the related problem of existence of measures of maximal entropy.