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

TL;DR

This paper proves mixing and equidistribution properties for discrete isometry groups of Hadamard spaces with rank one elements, extending classical results from negatively curved manifolds to more general non-positively curved spaces.

Contribution

It establishes mixing of Ricks' Bowen-Margulis measure and orbit equidistribution for groups with non-arithmetic length spectrum in Hadamard spaces.

Findings

Ricks' Bowen-Margulis measure is mixing under certain conditions.

Orbit points are equidistributed in the space.

Provides asymptotic estimates for orbit counting functions.

Abstract

Let $X$ be a proper, geodesically complete Hadamard space, and $\ \Gamma<\mbox{Is}(X)$ 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 $\Gamma$ has non-arithmetic 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 $\Gamma$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit counting function of $\Gamma$. This generalizes well-known facts for non-elementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT$(-1)$-spaces.