Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method

TL;DR

This paper establishes measure rigidity results for stationary measures under random walks and group actions, introducing the concept of generalized u-Gibbs states and demonstrating their invariance properties.

Contribution

It introduces the notion of generalized u-Gibbs states and proves their invariance, extending measure rigidity results to broader classes of dynamical systems.

Findings

Rigidity results for stationary measures of random walks on manifolds.

Invariance of generalized u-Gibbs states under certain dynamics.

Applicability to single diffeomorphisms and flows.

Abstract

We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or $1$-parameter flow and establishes extra invariance of a class of measures that we call ``generalized u-Gibbs states''.