Geodesic Flows Modelled by Expansive Flows

Gelfert, Katrin; Ruggiero, Rafael O.

doi:10.1017/s0013091518000160

Cited by 16 publications

(46 citation statements)

References 49 publications

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“…We remark that for surfaces without focal points, the uniqueness of the measures of maximal entropy was first proved by Gelfert and Ruggiero [GR19]. A recently work of Climenhaga, Knieper, and War [CKW19] further extended this result to geodesic flows over surfaces without conjugate points.…”

Section: Introductionmentioning

confidence: 81%

Unique equilibrium states for geodesic flows over surfaces without focal points

2020

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We prove that for closed rank 1 manifolds without focal points the equilibrium states are unique for Hölder potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property.Theorem A. Let M be a closed rank 1 Riemannian manifold without focal points, F be the geodesic flow over M , and ϕ : T 1 M → R be a Hölder potential. If ϕ satisfies the pressure gap condition, then ϕ has a unique equilibrium state µ ϕ .The origin of Theorem A traces back to the work of Bowen [Bow74] where he showed that the bounded distortion property (also known as the Bowen property) on the potential and the expansivity and the specification property on the dynamical system guarantee the existence of a unique equilibrium state. Natural examples of such systems are uniformly hyperbolic systems. Since the work of Bowen, his result has been extended in various directions, and one such direction aims at relaxing the assumptions on the base dynamical systems. Recently, Climenhaga and Thompson [CT16] developed a set of criteria that guarantees the existence of a unique equilibrium state which applies to many non-uniformly hyperbolic systems; see [CFT18], [BCFT18], [CKP18], and [CFT19].Among non-uniformly hyperbolic dynamical systems arising from geometry, the first result in the same flavor as Theorem A was the work of Knieper [Kni98]. He employed Patterson-Sullivan theory to establish the uniqueness of the measure of maximal entropy for geodesic flows over nonpositively curved manifolds. Twenty years later, Burns, Climenhaga, Fisher, and Thompson [BCFT18] extended Knieper's result to equilibrium states for potentials with the pressure gap. Their approach is inspired by the previously mentioned work of Bowen [Bow74]. Previous work by the authors [CKP18] used the same approach to further generalize this result to surfaces without focal points. Finally, in this work, Theorem A shows that the same philosophy holds for higher dimensional manifolds without focal points.From Theorem A, it automatically follows that these equilibrium states are ergodic. It is natural to ask whether such equilibrium states possess stronger ergodic properties. Indeed, for uniformly hyperbolic systems, the unique equilibrium states for Hölder potentials have many stronger ergodic properties; these include being Bernoulli, and having equidistribution property by weighted periodic orbits as well as statistical properties such as the central limit theorem and the large deviation property; see [PP90] and [KH97].The following theorem partially answers the question above, and establishes several ergodic properties of the unique equilibrium state µ ϕ from Theorem A.Theorem B. In the setting of Theorem A, the unique equilibrium state µ ϕ has the following properties: µ ϕ has the K-property and is fully supported. Moreover, µ ϕ is equal to the weak- * limit of the weig...

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Section: Introductionmentioning

confidence: 81%

Unique equilibrium states for geodesic flows over surfaces without focal points

2020

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“…The uniqueness referred in the previous paragraph follows from the work of Knieper, who proved it for geodesic flows on closed rank-one manifolds [Kni98], and also for geodesic flows on symmetric spaces of higher rank [Kni05]. Gelfert and Ruggiero proved the uniqueness for geodesic flows on surfaces without focal points and genus greater than one [GR19]. Burns et al proved the uniqueness of many equilibrium states (including some multiples of the geometric potential and the zero potential) of geodesic flows on rank-one manifolds [BCFT18], and there is a recent preprint that obtains similar results for geodesic flows on surfaces without focal points [CKP20].…”

Section: 1mentioning

confidence: 88%

Symbolic dynamics for non-uniformly hyperbolic systems

Lima¹

2020

Ergod. Th. Dynam. Sys.

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This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

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“…Local cross sections. Given a vector v ∈ T 1 M , let us construct a local cross section D(v) containing v (here we follow [17]). This section will be foliated by projections of leaves of the foliation W u (recall Section 2.5).…”

Section: 1mentioning

confidence: 99%