Pressure at infinity on countable Markov shifts

TL;DR

This paper investigates the pressure at infinity for potentials on countable Markov shifts, establishing semi-continuity results and criteria for equilibrium states, with extensions to suspension flows.

Contribution

It introduces new upper semi-continuity results for pressure at infinity and provides criteria for existence of equilibrium and maximizing measures.

Findings

Pressure at infinity is upper semi-continuous under certain conditions.

Criteria for existence of equilibrium states are established.

Results extend to suspension flows over countable Markov shifts.

Abstract

In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.