Rigidity properties for commuting automorphisms on tori and solenoids

Assuming positive entropy, we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for these actions.

Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $G$, we give a classification of the probability measures on $G/\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

Equidistribution of divergent orbits of the diagonal group in the space of lattices

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: the discriminant—an integer—and the type—an integer vector. We then study the question of the limit distributional behavior of these orbits as the discriminant goes to infinity. Using entropy methods we prove that, for divergent orbits of a specific type, virtually any sequence of orbits equidistributes as the discriminant goes to infinity. Using measure rigidity for higher-rank diagonal actions, we complement this result and show that, in dimension three or higher, only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.