Mathematical Research Letters

Volume 27 (2020)

Number 4

Upper semi-continuity of entropy in non-compact settings

Pages: 1055 – 1077



Godofredo Iommi (Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Santiago, Chile)

Mike Todd (School of Mathematics and Statistics, University of St Andrews, Mathematical Institute, St Andrews, Scotland, United Kingdom)

Aníbal Velozo (Department of Mathematics, Yale University, New Haven, Connecticut, U.S.A.)


We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous on the set of ergodic measures. Note that the phase space is non-compact. We also discuss the related problem of existence of measures of maximal entropy.

Received 12 October 2018

Accepted 28 February 2019

Published 14 December 2020