Differentiability of the pressure in non-compact spaces
Volume 259 / 2022
Fundamenta Mathematicae 259 (2022), 151-177
MSC: Primary 37D35; Secondary 28D20, 46T20.
DOI: 10.4064/fm182-3-2022
Published online: 28 July 2022
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.
