Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

Details Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary

2004-05-27

arxiv.org/abs/math/0405515v1

Mathematics

Dynamical Systems

Scientific paper

Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(y\gamma,b\gamma^{-1}):\gamma\in\Gamma} in X\times X(\infty). It is proved, in particular, that the orbits of \Gamma in the Furstenberg boundary are equidistributed, and that the orbits of \Gamma in X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sullivan theory. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces.

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