  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.
