Effective density of unipotent orbits
Advances in Homogeneous Dynamics May 11, 2015 - May 15, 2015
Location: MSRI: Simons Auditorium
dynamics on Lie groups
37-XX - Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
37D40 - Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37Cxx - Smooth dynamical systems: general theory [See also 34Cxx, 34Dxx]
37C40 - Smooth ergodic theory, invariant measures [See also 37Dxx]
Raghunathan conjectured that If G is a Lie group, Gamma a lattice, p in G/Gamma, and U an (ad-)unipotent group then the closure of U.p is homogeneous (a periodic orbit of a subgroup of G). This conjecture was proved by Ratner in the early 90's via the classification of invariant measures; significant special cases were proved earlier by Dani and Margulis using a different, topological dynamics approach. Neither of these proofs is effective, nor do they provide rates --- e.g. if p is generic in the sense that it does not lie on a
periodic orbit of any proper subgroup L<G with U<=L, an estimate (possibly depending on diophantine-type properties of the pair (p,U))) how large a piece of an orbit is needed so that it comes within distance epsilon of any point in a given compact subset of G/Gamma
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