On the Vere-Jones classi cation and existence of maximal measures for topological Markov chains

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. Introduction In this article we are interested in connected oriented graphs and topological Markov chains. All the graphs we consider have a countable set of vertices. If G is an oriented graph, let G be the set of two-sided in nite paths in G and let denote the shift transformation. The Markov chain associated to G is the (non compact) dynamical system ( G; ). The entropy h(G) of the Markov chain G was de ned by Gurevich; it can be computed by several ways and satis es the Variational Principle [7],[8]. In [14] Vere-Jones classi es connected oriented graphs as transient, null recurrent or positive recurrent according to the properties of the series associated with the number of loops. To a certain extend, positive recurrent graphs resemble nite graphs. In [7] Gurevich shows that a Markov chain admits a measure of maximal entropy (also called maximal measure) if and only if the graph is positive recurrent. In this case, this measure is unique and it is an ergodic Markov measure. When the graph is positive recurrent, its incidence matrix admits a maximal eigenvalue and the associated left and right eigenvectors allow to de ne a probability Markov measure which is the unique maximal measure; this construction is the same as in the nite case. In [11] Salama gives a geometric approach to the Vere-Jones classi cation. The fact that a graph can (or cannot) be \extended" or \contracted" without changing its entropy is closely related to its class. Salama rst thought that these properties entirely determine whether a graph is transient, null recurrent of positive recurrent. In particular he stated that a graph is positive recurrent if and only if it has no proper subgraph of equal entropy but he soon realized that the proof of the \only if" part was false (see Theorem 2.3 and Errata in [11]). The result itself is not valid: in [5] Fiebig gives an example of a locally compact positive recurrent graph without any subgraph of equal entropy. This result shows that the positive recurrent class splits into two subclasses: a graph is called strongly positive recurrent if it has no proper subgraph of equal entropy; it is equivalent to a combinatorial condition (a nite connected graph always satis es this property). In [11] Salama also states that a graphs is transient if and only if it can be extended to a bigger transient graph of equal entropy. We show that any transient graph G is contained in a recurrent graph of equal entropy, which is positive or null recurrent depending on the properties of G. We illustrate the two possibilities { a transient graph with a positive or null recurrent extension { by an example. It also