Recurrence and transience for suspension flows

Abstract

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.

Entropy in the cusp and phase transitions for geodesic flows

“…It is possible to show (see [Sa1, Theorem 1]) that the limit always exists and that it does not depend on i 1 . The following two properties of the pressure will be relevant for our purposes (see [Sa1,Theorems 2 and 3] and [IJT,Theorem 2.10]). If ϕ : Σ`Ñ R is a function of summable variations, then…”

“…It was proved in [IJT,Theorem 3.5] that potentials f for which ∆ f is locally Hölder have at most one equilibrium measure. Moreover, the following result (see [BI,Theorem 4]) characterises functions having equilibrium measures.…”

Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature

“…In [17,18], in a symbolic context, Sarig established two finiteness criteria for the measure associated with a Hölder potential F . Iommi and its collaborators extended this study to suspension flows over such shifts, in [4,11]. In the case of geodesic flows on the unit tangent bundle of noncompact manifolds, the first criterion appeared in [8].…”

Measures of maximal entropy for suspension flows over the full shift

“…Examples of suspension flows with phase transitions at the zero potential (i.e. examples for which there does not exist a unique MME) have previously been obtained when the alphabet is infinite by Iommi, Jordan and Todd [10,11], and when the roof function is allowed to have zeroes by Savchenko [19]. In these examples, the phase transition occurs because of non-existence of an MME rather than non-uniqueness.…”

“…In this case, the flow is expansive so the existence of MME is guaranteed [1]. Phase transitions must therefore arise from the presence of multiple MME rather than the mechanisms of [11,19].…”