Generalized Patterson-Sullivan measures for products of Hadamard spaces

TL;DR

This paper extends Patterson-Sullivan measures to product Hadamard spaces with specific group actions, analyzing their properties and relating Hausdorff dimension to exponential growth rates of the group orbits.

Contribution

It generalizes the classical Patterson-Sullivan construction to invariant subsets of the limit set in product Hadamard spaces, providing measure-theoretic insights.

Findings

Constructed conformal densities supported on invariant subsets.

Established bounds on Hausdorff dimension of the radial limit set.

Linked Hausdorff dimension to exponential growth rates.

Abstract

Let $\Gamma$ be a discrete group acting by isometries on a product $X=X_1\times X_2$ of Hadamard spaces. We further require that $X_1$, $X_2$ are locally compact and $\Gamma$ contains two elements projecting to a pair of independent rank one isometries in each factor. Apart from discrete groups acting by isometries on a product of CAT(-1)-spaces, the probably most interesting examples of such groups are Kac-Moody groups over finite fields acting on the Davis complex of their associated twin building. In a previous article we showed that the regular geometric limit set $\Lim$ splits as a product $F_\Gamma\times P_\Gamma$, where $F_\Gamma\subseteq\rand_1\times \rand_2$ is the projection of the geometric limit set to $\rand_1\times \rand_2$, and $P_\Gamma$ encodes the ratios of the speed of divergence of orbit points in each factor. Our aim in this paper is a description of the limit set from a measure theoretical point of view. We first study the conformal density obtained from the classical Patterson-Sullivan construction, then generalize this construction to obtain measures supported in each $\Gamma$-invariant subset of the regular limit set and investigate their properties. Finally we show that the Hausdorff dimension of the radial limit set in each $\Gamma$-invariant subset of $\Lim$ is bounded above by the exponential growth rate introduced in the previous article.