Nonuniform measure rigidity

TL;DR

This paper proves that for certain higher-rank smooth group actions on manifolds, hyperbolic invariant measures with positive entropy are necessarily absolutely continuous, extending measure rigidity results.

Contribution

It establishes absolute continuity of invariant measures under general conditions for higher-rank abelian group actions, broadening previous measure rigidity theorems.

Findings

Hyperbolic measures with positive entropy are absolutely continuous.

Absolute continuity of conditional measures on Lyapunov foliations is proven.

Results apply to smooth actions of Z^k and R^k with hyperbolic measures.

Abstract

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element of the action has positive entropy, then $\mu$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.