Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups

TL;DR

This paper establishes absolute continuity and rigidity results for certain non-uniformly hyperbolic actions of higher rank abelian groups, focusing on Lyapunov exponents and cocycle behavior.

Contribution

It proves absolute continuity of high entropy measures under specific conditions and demonstrates cocycle rigidity for actions on tori and infranilmanifolds.

Findings

Absolute continuity of high entropy measures for certain actions

Rigidity results for cocycles over these actions

Existence of absolutely continuous invariant measures in specific cases

Abstract

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.