Escape of mass and entropy for diagonal flows in real rank one situations

Einsiedler, Manfred; Kadyrov, Shirali; Pohl, Anke

doi:10.1007/s11856-015-1252-y

Cited by 20 publications

(14 citation statements)

References 14 publications

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“…In particular, similar results for the geodesic flow on the modular surface were proved by Einsiedler, Lindenstrauss, Michel and Venkatesh in [12]. Einsiedler, Kadyrov and Pohl generalized these results to diagonal actions on spaces Γ\G where G is a connected semisimple real Lie group of rank 1 with finite center, and Γ is a lattice [11]. Finally, Iommi, Riquelme and Velozo (in two papers with different sets of coauthors) considered entropy in the cusp for geometrically finite Riemannian manifolds with pinched negative sectional curvature and uniformly bounded derivatives of the sectional curvature [18,28].…”

Section: Thensupporting

confidence: 59%

Excursions to the cusps for geometrically finite hyperbolic orbifolds, and equidistribution of closed geodesics in regular covers

Mor¹

2020

Preprint

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We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion we prove new results on the distribution of collections of closed geodesics on such orbifold, and as a corollary we prove equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.

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Section: Thensupporting

confidence: 59%

Excursions to the cusps for geometrically finite hyperbolic orbifolds, and equidistribution of closed geodesics in regular covers

Mor¹

2020

Preprint

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“…In this generality, it is not known if the Hausdorff dimension of the set (1) is strictly less than dim G. This was observed in [12] for X = G/Γ when G is of R-rank one.…”

Section: Introductionmentioning

confidence: 99%

Exceptional sets in homogeneous spaces and Hausdorff dimension

Kadyrov

2014

Dynamical Systems

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In this paper, we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces X to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius r is at mostwhere C > 0 is a constant independent of r > 0. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.

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“…Note that another definition of entropy at infinity appears in [BBG14, RV19,Vel17], which is somehow the maximal entropy of a sequence of invariant probability measures diverging to infinity. See also [EKP15,ELMV12] for related works in finite volume rank one homogeneous dynamics. It follows from [RV19] in the geometrically finite case and [Vel17] more generally that this entropy at infinity coincides with our definition.…”

Section: Entropy At Infinity Spr Manifolds and Bowen-margulis Measuresmentioning

confidence: 99%

Regularity of entropy, geodesic currents and entropy at infinity

Schapira¹,

Tapie²

2021

ASENS

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In this work, we introduce a notion of entropy at infinity for the geodesic flow of negatively curved manifolds. We introduce the class of noncompact manifolds which admit a critical gap between entropy at infinity and topological entropy. We call them strongly positively recurrent manifolds (SPR), and provide many examples. We show that dynamically, they behave as compact manifolds. In particular, they admit a finite measure of maximal entropy.Using the point of view of currents at infinity, we show that on these SPR manifolds the topological entropy of the geodesic flow varies in a C 1 -way along (uniformly) C 1 -perturbations of the metric. This result generalizes former work of Katok (1982) and Katok-Knieper-Weiss (1991) in the compact case. RésuméDans ce travail, nous introduisons une notion d'entropie à l'infini pour les flots géodésiques des variétés à courbure négative. Nous introduisons la classe des variétés, dites fortement positivement récurrentes (SPR), dont l'entropie à l'infini est strictement inférieure à l'entropie topologique. Nous donnons de nombreux exemples de telles variétés. Nous montrons que d'un point de vue dynamique, ces variétés ressemblent à des variétés compactes. En particulier, elles admettent une mesure finie maximisant l'entropie.A l'aide du point de vue des courants à l'infini, nous montrons que sur ces variétés SPR, l'entropie topologique varie de manière C 1 le long de perturbations C 1 uniformes de la métrique. Ceci généralise des résultats passés de Katok (1982) et Katok-Knieper-Weiss (1991 dans le cas compact.

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Escape of mass and entropy for diagonal flows in real rank one situations

Cited by 20 publications

References 14 publications

Excursions to the cusps for geometrically finite hyperbolic orbifolds, and equidistribution of closed geodesics in regular covers

Excursions to the cusps for geometrically finite hyperbolic orbifolds, and equidistribution of closed geodesics in regular covers

Exceptional sets in homogeneous spaces and Hausdorff dimension

Regularity of entropy, geodesic currents and entropy at infinity

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