Escape of entropy for countable Markov shifts

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 mass, measure theoretic entropy, and entropy at infinity system. This relation has several consequences. For example obtain that map is upper semi-continuous measures form an dense subset. results also provide new proofs describing existence stability maximal entropy. We relate with Hausdorff dimension set recurrent points on average. Of independent interest, prove a version Katok's formula in setting.