Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups (Q1889814)

Let \(X\) be a symmetric space of noncompact type, \(G=\text{Isom}^{\circ}(X)\) and \(\Gamma \subset G\) a discrete subgroup. Considering an appropriate Hausdorff measure on the geometric boundary \(\partial X\) the author proves that for regular boundary points \(\xi \in {\partial X}\), the Hausdorff dimension of the radial limit set in \(G\bullet\xi\) is bounded above by the exponential growth rate of the number of orbit points close in direction to \(G\bullet\xi\subseteq {\partial X}\). Furthermore, for Zariski dense discrete groups \(\Gamma\), the author constructs \(\Gamma\)-invariant densities with support in every \(G\)-invariant subset of the limit set and studies their properties.