Transitive topological Markov chains of given entropy and period with or without measure of maximal entropy

Sylvie Ruette

Vol. 303 (2019), No. 1, 317–323

DOI: 10.2140/pjm.2019.303.317

Abstract

We show that, for every positive real number $h$ and every positive integer $p$, there exist oriented graphs $G,G^{\prime }$ (with countably many vertices) that are strongly connected, of period $p$, of Gurevich entropy $h$, and such that $G$ is positive recurrent (thus the topological Markov chain on $G$ admits a measure of maximal entropy) and $G^{\prime }$ is transient (thus the topological Markov chain on $G^{\prime }$ admits no measure of maximal entropy).

Keywords

topological Markov chain, countable oriented graph, topological entropy

Mathematical Subject Classification 2010

Primary: 37B10

Secondary: 37B40

Milestones

Received: 26 June 2018

Accepted: 8 July 2019

Published: 21 December 2019