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

Abstract

Let $X$ be a proper, geodesically complete Hadamard space, and $\mathrm{\Gamma }<Is\left(X\right)$ 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 $\mathrm{\Gamma }$ has nonarithmetic length spectrum, then the Ricks–Bowen–Margulis measure — which generalizes the well-known Bowen–Margulis measure in the CAT$\left(-1\right)$ setting — is mixing. If in addition the Ricks–Bowen–Margulis measure is finite, then we also have equidistribution of $\mathrm{\Gamma }$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit counting function of $\mathrm{\Gamma }$. This generalizes well-known facts for nonelementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT$\left(-1\right)$-spaces.