Expanding measures: Random walks and rigidity on homogeneous spaces

Abstract

Let G be a real Lie group, Λ<G a lattice and H<G a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures μ on H and, applying recent work of Eskin-Lindenstrauss, prove that μ-stationary probability measures on G/Λ are homogeneous. Transferring a construction by Benoist-Quint and drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on G/Λ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in G/Λ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a non-conformal and weighted setting.