Recurrence and transience for suspension flows

TL;DR

This paper investigates the thermodynamic formalism for suspension flows over countable Markov shifts, focusing on recurrence, transience, and equilibrium measures, with applications to interval maps with parabolic fixed points.

Contribution

It introduces conditions for existence and uniqueness of equilibrium measures and defines the renewal flow to model diverse recurrence behaviors.

Findings

Established criteria for equilibrium measure existence and uniqueness.

Defined recurrence and transience notions for these flows.

Analyzed phase transitions in the thermodynamic formalism.

Abstract

We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the "renewal flow", which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.