Differentiability of the pressure in non-compact spaces

TL;DR

This paper investigates the differentiability of pressure in non-compact dynamical systems, especially countable Markov shifts, by using metric compactifications to analyze regularity and phase transitions.

Contribution

It introduces a method to prove pressure differentiability on residual sets in non-compact spaces and establishes a criterion linking original systems to their compactifications.

Findings

Pressure is differentiable on a residual set in non-compact spaces.

The sectorially arranged property ensures pressure equivalence between original and compactified systems.

Examples demonstrate compactifications with complex boundary structures like Cantor sets.

Abstract

Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so called sectorially arranged property, which implies that the pressure in the original system and in the compactification coincide. Examples showing that the compactifications can have rich boundaries, for example a Cantor set, are provided.