# On the Vere-Jones classification and existence of maximal measures for countable topological Markov chains

## TL;DR

This paper investigates the classification of countable topological Markov chains, demonstrating how transient graphs can be extended to recurrent ones with equal entropy, and establishing conditions for the existence of maximal measures based on local entropy.

## Contribution

It extends Vere-Jones classification to countable Markov chains and introduces local entropy to determine the existence of maximal measures.

## Key findings

- Transient graphs can be extended to recurrent graphs with the same entropy.
- Examples of positive and null recurrent graphs are provided.
- A transitive Markov chain admits a maximal measure if its local entropy is less than its global entropy.

## Abstract

We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maximal entropy (or maximal measure) whenever its local entropy is less than its (global) entropy.