Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry

Gabriele Link

doi:10.3934/dcds.2018245

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2018, Volume 38, Issue 11: 5577-5613. Doi: 10.3934/dcds.2018245

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Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry

Gabriele Link^,

Institut für Algebra und Geometrie, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76 131 Karlsruhe, Germany

* Corresponding author: Gabriele Link
* Corresponding author: Gabriele Link

Received: November 2017

Revised: June 2018

Published: August 2018

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Abstract

Let $X$ be a proper Hadamard space and $\Gamma <{\text{Is}}(X)$ a non-elementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient $\Gamma \backslash X$ and with respect to Ricks' measure introduced in [35]. This generalizes previous work of the author and J. C. Picaud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting and with respect to Knieper's measure.

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

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