Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical theory of normal forms for local contractions.Comment: 28 pages, LaTe

AN INVITATION TO RIGIDITY THEORY

This survey is dedicated to Professor Anatole Katok on the occasion of his sixtieth birthday. He has made numerous important contributions to dynamics and ergodic theory proper. During the last two decades, he was one of the key researchers in the field of rigidity or geometric rigidity. My goal here is to give a bird’s eye view of this subject with particular attention to the many ideas and topic

Invariant measures for higher-rank hyperbolic abelian actions

We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of Rk, Zk and Zk +. We show that they are either Haar measures or that every element of the action has zero metric entropy