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

Inhyeok Choi, Dongryul M. Kim

TL;DR

The paper develops a geometric, flow-free framework to classify Γ–invariant Radon measures on ML for non-elementary subgroups Γ of Mod(S), focusing on the recurrence locus and the dichotomy between divergence and convergence types. It constructs an explicit, Γ–invariant measure μ_Γ from a δ_Γ–dimensional conformal density, proves uniqueness on the recurrence locus for divergence-type Γ, and extends the analysis to CAT(−1) spaces via squeezing isometries. For convex cocompact subgroups, it yields a complete measure classification and an orbit-closure description, linking growth, boundary behavior, and horospherical dynamics through Patterson–Sullivan theory with squeezing. The approach generalizes known Mod(S) results and provides a unifying geometric method applicable to Teichmüller spaces and other partially hyperbolic metric spaces, with potential extensions to higher-rank and non-geodesic contexts.

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 Hamenstädt'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 Teichmüller spaces and partially $\operatorname{CAT}(-1)$ spaces.

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

TL;DR

The paper develops a geometric, flow-free framework to classify Γ–invariant Radon measures on ML for non-elementary subgroups Γ of Mod(S), focusing on the recurrence locus and the dichotomy between divergence and convergence types. It constructs an explicit, Γ–invariant measure μ_Γ from a δ_Γ–dimensional conformal density, proves uniqueness on the recurrence locus for divergence-type Γ, and extends the analysis to CAT(−1) spaces via squeezing isometries. For convex cocompact subgroups, it yields a complete measure classification and an orbit-closure description, linking growth, boundary behavior, and horospherical dynamics through Patterson–Sullivan theory with squeezing. The approach generalizes known Mod(S) results and provides a unifying geometric method applicable to Teichmüller spaces and other partially hyperbolic metric spaces, with potential extensions to higher-rank and non-geodesic contexts.

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 Hamenstädt'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 Teichmüller spaces and partially spaces.
Paper Structure (46 sections, 36 theorems, 228 equations, 2 figures)

This paper contains 46 sections, 36 theorems, 228 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Measure classification for full $\operatorname{Mod}(S)$: finite covolume
  3. Measure classification for subgroups: infinite covolume
  4. Main statements
  5. Explicit construction of invariant Radon measures
  6. Horospherical foliations of $\operatorname{CAT}(-1)$ spaces
  7. More examples
  8. Metric spaces with squeezing isometries
  9. On the proof
  10. Non-geodesic spaces
  11. Organization
  12. Teichmüller theory
  13. Thurston boundary
  14. Gardiner--Masur boundary
  15. Uniquley ergodic laminations and boundary comparison
  16. ...and 31 more sections

Key Result

Theorem 1.1

Let $\mu$ be a $\operatorname{Mod}(S)$-invariant Radon measure on $\mathcal{R}_{\operatorname{Mod}(S)}$. Then $\mu$ is a constant multiple of $\mu_{\rm Th}$.

Figures (2)

  • Figure 1: A contracting geodesic (left) and a squeezing geodesic (right)
  • Figure 2: Alignment of geodesics and points.

Theorems & Definitions (81)

  • Theorem 1.1: lindenstrauss2008ergodic, hamenstadt2009invariant
  • Theorem 1.2: Ergodicity
  • Theorem 1.3: Unique ergodicity
  • Example 1.4: Divergence-type subgroups
  • Theorem 1.5: Convex cocompact subgroups
  • Remark 1.6
  • Theorem 1.7: $\operatorname{CAT}(-1)$ spaces
  • Theorem 1.8: Ergodicity
  • Remark 1.9
  • Theorem 1.10: Unique ergodicity
  • ...and 71 more