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

TL;DR

This paper demonstrates the existence of strongly connected topological Markov chains with specified entropy and period, differentiating between those that admit a measure of maximal entropy and those that do not, based on recurrence properties.

Contribution

It constructs explicit examples of Markov chains with given entropy and period, distinguishing between positive recurrence and transience in the context of maximal entropy measures.

Findings

Existence of graphs with prescribed entropy and period

Construction of positive recurrent graphs with maximal entropy measures

Construction of transient graphs without maximal entropy measures

Abstract

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