Ergodic geometry for non-elementary rank one manifolds

2015/08/24 by Gabriele Link, Link, Gabriele, Jean-Claude Picaud +1 · 1 citation
Mathematics · #22D40 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1508.05854

openalex publication_date 2015/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a Hadamard manifold, and Γ a non-elementary discrete group 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/Γ to the behavior of the Poincaré series of Γ. 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 Γ-invariant conformal densities supported on the geometric limit set of Γ.

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• On the almost sure spiraling of geodesics in CAT(0) spaces

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