#### Vol. 303, No. 1, 2019

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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
Primary: 37B10
Secondary: 37B40
##### Milestones
Received: 26 June 2018
Accepted: 8 July 2019
Published: 21 December 2019
##### Authors
 Sylvie Ruette Laboratoire de Mathématiques d’Orsay, UMR 8628 Université Paris-Saclay Orsay France