Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact

Abstract: To study the asymptotic behavior of orbits of a dynamical system, one can look at orbit closures or invariant measures. When the underlying system has a homogeneous structure, usually coming from a Lie group, with appropriate assumptions a wide range of rigidity theorems show that ergodic invariant measures and orbit closures have to be well-behaved and can often be classified.

I will describe joint work with Brown, Eskin, and Rodriguez-Hertz, which establishes rigidity results for quite general dynamical systems having some hyperbolicity. I will also explain some of the necessary assumptions as well as the homogeneous structures that emerge. Time permitting, I will also explain closely related joint work with Fisher and Lowe on geodesic flows in negative curvature.