In this talk, we discuss geometric group theoretic approaches to classifying invariant Radon measures on various dynamical systems arising from Teichmüller dynamics and homogeneous dynamics.

First, given a non-elementary subgroup of the mapping class group of a hyperbolic surface, we construct an invariant Radon measure on the space of measured laminations and prove that any ergodic invariant Radon measure on the recurrent measured laminations coincides with this measure. As a special case, we deduce that for a convex cocompact subgroup, any ergodic invariant Radon measure is either the unique one supported on the recurrent measured laminations, or a counting measure on the orbit of a non-recurrent measured lamination. When the subgroup is the full mapping class group, this measure is the Thurston measure, and in this case the uniqueness on recurrent measured laminations was independently proved by Lindenstrauss--Mirzakhani and Hamenstädt.

Next, we classify all horospherical-invariant Radon measures on higher-rank Anosov and relatively Anosov homogeneous spaces. These results are joint work with Inhyeok Choi.