  In this paper we study ergodic theory of countable Markov shifts. These are
dynamical systems defined over non-compact spaces. Our main result relates the
escape of mass, the measure theoretic entropy, and the entropy at infinity of
the system. This relation has several consequences. For example we obtain that
the entropy map is upper semi-continuous and that the ergodic measures form an
entropy dense subset. Our results also provide new proofs of results describing
the existence and stability of the measure of maximal entropy. We relate the
entropy at infinity with the Hausdorff dimension of the set of recurrent points
that escape on average. Of independent interest, we prove a version of Katok's
entropy formula in this non-compact setting.
