Central limit theorem for the geodesic flow associated with a Kleinian group, case δ > d/2

N. Enriquez; J. Franchi; Y. Le Jan

doi:10.1016/S0021-7824(00)01182-X

Central limit theorem for the geodesic flow associated with a Kleinian group, case δ > d/2

N. Enriquez, J. Franchi, Y. Le Jan

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let Γ be a geometrically finite Kleinian group, relative to the hyperbolic space ℍ = ℍd+1, and let δ denote the Hausdorff dimension of its limit set, that we suppose here strictly larger than d/2. We prove a central limit theorem for the geodesic flow on the manifold M := Γ\ℍ, with respect to the Patterson-Sullivan measure. The argument uses the ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a potential operator is proved.

Original language	English (US)
Pages (from-to)	153-175
Number of pages	23
Journal	Journal des Mathematiques Pures et Appliquees
Volume	80
Issue number	2
DOIs	https://doi.org/10.1016/S0021-7824(00)01182-X
State	Published - Mar 2001

Keywords

Central limit theorem
Diffusion process
Geodesic flow
Hyperbolic manifold of infinite volume
Patterson-Sullivan measure
Spectral gap
Stable foliation

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1016/S0021-7824(00)01182-X

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title = "Central limit theorem for the geodesic flow associated with a Kleinian group, case δ > d/2",

abstract = "Let Γ be a geometrically finite Kleinian group, relative to the hyperbolic space ℍ = ℍd+1, and let δ denote the Hausdorff dimension of its limit set, that we suppose here strictly larger than d/2. We prove a central limit theorem for the geodesic flow on the manifold M := Γ\ℍ, with respect to the Patterson-Sullivan measure. The argument uses the ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a potential operator is proved.",

keywords = "Central limit theorem, Diffusion process, Geodesic flow, Hyperbolic manifold of infinite volume, Patterson-Sullivan measure, Spectral gap, Stable foliation",

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AU - Le Jan, Y.

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N2 - Let Γ be a geometrically finite Kleinian group, relative to the hyperbolic space ℍ = ℍd+1, and let δ denote the Hausdorff dimension of its limit set, that we suppose here strictly larger than d/2. We prove a central limit theorem for the geodesic flow on the manifold M := Γ\ℍ, with respect to the Patterson-Sullivan measure. The argument uses the ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a potential operator is proved.

AB - Let Γ be a geometrically finite Kleinian group, relative to the hyperbolic space ℍ = ℍd+1, and let δ denote the Hausdorff dimension of its limit set, that we suppose here strictly larger than d/2. We prove a central limit theorem for the geodesic flow on the manifold M := Γ\ℍ, with respect to the Patterson-Sullivan measure. The argument uses the ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a potential operator is proved.

KW - Central limit theorem

KW - Diffusion process

KW - Geodesic flow

KW - Hyperbolic manifold of infinite volume

KW - Patterson-Sullivan measure

KW - Spectral gap

KW - Stable foliation

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Central limit theorem for the geodesic flow associated with a Kleinian group, case δ > d/2

Abstract

Keywords

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