Expanding measures: Random walks and rigidity on homogeneous spaces

Abstract

Let $G$ be a real Lie group, $\Lambda <G$ a lattice and $H<G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of $H$-expanding measures $\mu$ on $H$ and, applying recent work of Eskin-Lindenstrauss, prove that $\mu$-stationary probability measures on $G/\Lambda$ 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 $G/\Lambda$ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\Lambda$ 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.

63 pages; revised the presentation of the proof of Corollary 1.2 and made other small changes and corrections. Accepted for publication by Forum of Mathematics, Sigma

References in corpus (4)

• Dimension and measures on sub-self-affine sets
• Spread Out Random Walks on Homogeneous Spaces
• Affine random walks on the torus
• Equidistribution of affine random walks on some nilmanifolds

Cited by in corpus (2)

• Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation
• Dimension bounds for singular affine forms