Vol. 2, No. 4, 2020

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Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

Gabriele Link

Vol. 2 (2020), No. 4, 791–839
Abstract

Let X be a proper, geodesically complete Hadamard space, and Γ < Is(X) a discrete subgroup of isometries of X with the fixed point of a rank one isometry of X in its infinite limit set. In this paper we prove that if Γ has nonarithmetic length spectrum, then the Ricks–Bowen–Margulis measure — which generalizes the well-known Bowen–Margulis measure in the CAT(1) setting — is mixing. If in addition the Ricks–Bowen–Margulis measure is finite, then we also have equidistribution of Γ-orbit points in X, which in particular yields an asymptotic estimate for the orbit counting function of Γ. This generalizes well-known facts for nonelementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT(1)-spaces.

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Keywords
rank one space, Bowen–Margulis measure, mixing, equidistribution, orbit counting function
Mathematical Subject Classification 2010
Primary: 20F69, 22D40
Secondary: 20F67, 37D25, 37D40
Milestones
Received: 25 October 2018
Revised: 17 July 2019
Accepted: 11 August 2019
Published: 10 December 2019
Authors
Gabriele Link
Institute for Algebra and Geometry
Karlsruhe Institute of Technology (KIT)
Karlsruhe
Germany