Invariant measures on the space of measured laminations for subgroups of mapping class group

TL;DR

This paper classifies invariant measures on the space of measured laminations for certain subgroups of the mapping class group, extending previous results and providing a geometric, flow-independent approach applicable to various metric spaces.

Contribution

It generalizes known ergodic measure classifications to broader subgroups and introduces a geometric method that avoids reliance on flows or homogeneous dynamics.

Findings

Classifies invariant Radon measures for non-elementary subgroups.

Shows unique ergodicity for divergence-type subgroups with explicit measure construction.

Identifies invariant measures for convex cocompact subgroups as either unique or counting measures.

Abstract

For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a divergence-type subgroup, we show the uniquely ergodic by explicitly constructing the ergodic measure. This generalizes Lindenstrauss--Mirzakhani's result and Hamenst\"adt's result for the full mapping class group, in which case the ergodic measure is the Thurston measure. As a special case, we deduce that for a convex cocompact subgroup, every invariant ergodic Radon measure on the space of all measured laminations is either the unique measure on recurrent measured laminations, or a counting measure on the orbit of a non-recurrent measured lamination. Our method is geometric and does not rely on continuous or homogeneous flows on the ambient space or a dynamical system associated with a finite measure space. This leads to a unifying approach for various metric spaces, including Teichm\"uller spaces and partially $\operatorname{CAT}(-1)$ spaces.