Expanding measures and random walks on homogeneous spaces

Abstract: We will start by reviewing some recent works on random walks on homogeneous spaces. We will continue by discussing the notion of a H-expanding probability measure on a connected semisimple Lie group H, that we introduce inspired by these developments. As we shall see, for a H-expanding µ with H < G, on the one hand, one can obtain a description of µ-stationary probability measures on the homogeneous space G/Λ using the measure classification results of Eskin– Lindenstrauss, and on the other hand, the recurrence techniques of Benoist–Quint can be generalized to this setting. As a result, we will deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariski-dense subgroups and some epimorphic subgroups of H. If time allows, we will see how, using an idea of Simmons–Weiss, this allows also us to deduce Birkhoff genericity of a class of fractal measures with respect to expanding diagonal actions. Joint work with Roland Prohaska and Ronggang Shi.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar