Generalized Patterson-Sullivan measures for products of Hadamard spaces

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.