Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

Abstract.

Let X = G/K be a Riemannian symmetric space of noncompact type and $\Gamma$ a discrete “generic” subgroup of G with critical exponent $\delta(\Gamma)$. Denote by $X_{reg} (\infty)$ the set of regular elements of the geometric boundary $X(\infty)$ of X. We show that the support of all $\Gamma$-invariant conformal densities of dimension $\delta(\Gamma)$ on $X_{reg} (\infty)$ (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on $X(\infty)$. This provides information on the large-scale growth of $\Gamma$-orbits in X. If in addition we assume $\Gamma$ to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine $\delta(\Gamma)$ and this G-orbit for lattices, and show that they are of divergence type.