Expanding measures: Random walks and rigidity on homogeneous spaces

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 nonconformal and weighted setting.