Suspension Flows Over Countable Markov Shifts

Barreira, Luís; Iommi, Godofredo

doi:10.1007/s10955-006-9140-9

Cited by 29 publications

(35 citation statements)

References 17 publications

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“…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.…”

Section: Suspension Flowsmentioning

confidence: 99%

Entropy in the cusp and phase transitions for geodesic flows

Iommi¹,

Riquelme²,

Velozo³

2018

Isr. J. Math.

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In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one side on symbolic methods and Markov partitions and on the other on geometric techniques and approximation properties at level of groups.

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Section: Suspension Flowsmentioning

confidence: 99%

Entropy in the cusp and phase transitions for geodesic flows

Iommi¹,

Riquelme²,

Velozo³

2018

Isr. J. Math.

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“…Lemma 3.3 [Kem11]), this definition is independent with the choice of a ∈ S. Moreover, there are several alternative ways of defining the Gurevich pressure for suspension flows such as using the variational principle. In the following, we summarize some of them from works of Savchenko [Sav98], Barreira-Iommi [BI06], Kempton [Kem11], and Jaerisch-Kesseböhmer-Lamei [JKL14].…”

Section: 2mentioning

confidence: 99%

Manhattan curves for hyperbolic surfaces with cusps

Kao¹

2018

Ergod. Th. Dynam. Sys.

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In this paper, we study an interesting curve, so-called the Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface, especially representations corresponding to Riemann surfaces with cusps. Using Thermodynamic Formalism (for countable Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Marc Burger [Bur93] and Richard Sharp [Sha98] for convex-cocompact Fuchsian representations.

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“…The variational principle has been proved in the context of suspension flows defined over countable Markov shifts (analogous to Theorem 2.1) with different degrees of generality (see [BI1,JKL,Ke,Sav]). The version we will be interested here is the following:…”

Section: Existence and Uniqueness Of Equilibrium Measuresmentioning

confidence: 99%

Recurrence and transience for suspension flows

2015

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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.

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Suspension Flows Over Countable Markov Shifts

Cited by 29 publications

References 17 publications

Entropy in the cusp and phase transitions for geodesic flows

Entropy in the cusp and phase transitions for geodesic flows

Manhattan curves for hyperbolic surfaces with cusps

Recurrence and transience for suspension flows

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