Ergodic geometry for non-elementary rank one manifolds

Gabriele  Link; Jean-Claude  Picaud

doi:10.3934/dcds.2016072

Article Contents

2016, Volume 36, Issue 11: 6257-6284. Doi: 10.3934/dcds.2016072

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Ergodic geometry for non-elementary rank one manifolds

Gabriele Link^1, and
Jean-Claude Picaud^2,

1.
Institut für Algebra und Geometrie, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76 131 Karlsruhe
2.
LMPT, UMR 6083, Faculté des Sciences et Techniques, Parc de Grandmont, 37 200 Tours

Received: February 2013

Revised: May 2016

Published online: August 2016

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Abstract

Let $X$ be a Hadamard manifold, and $\Gamma\subset Is(X)$ a non-elementary discrete subgroup of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of the Poincaré series of $\Gamma$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $\Gamma$-invariant conformal densities supported on the geometric limit set of $\Gamma$.

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Mathematics Subject Classification: Primary: 37D40, 28D20; Secondary: 37D25, 20F67.

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