Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity

In this paper we introduce Patterson–Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson–Sullivan measure. For such systems we prove a generalization of Tukia’s measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson–Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia’s theorem for word hyperbolic groups, Teichmüller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets.