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Periodic approximation of topological Lyapunov exponents and the joint spectral radius for cocycles of mapping classes of surfaces

Anders Karlsson, Reza Mohammadpour

TL;DR

This work extends periodic-approximation results for leading Lyapunov exponents from linear cocycles to cocycles valued in the mapping class group by exploiting a closing property for the base dynamics. It introduces a metric joint spectral radius in the MC(G) setting, realized by ergodic measures along curves in the surface, and shows Lyapunov growth can be captured by periodic orbits and by subadditive growth of Teichmüller/Thurston distances. The authors combine ergodic theory, Teichmüller geometry, and spectral theory to connect Lyapunov exponents with joint spectral-radius growth, including a thermodynamic formalism for subadditive potentials. The results suggest a broad framework for similar phenomena in non-linear group actions on geometric spaces and hint at extensions to other metrics and groups, such as Kobayashi or Hofer geometries.

Abstract

We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be approximated by periodic orbits. We also extend the notion of the joint spectral radius to this setting, interpreting it via the exponential growth of curves under iterated mapping classes. Our approach connects ideas from ergodic theory, Teichmüller geometry, and spectral theory, and suggests a broader framework for similar results.

Periodic approximation of topological Lyapunov exponents and the joint spectral radius for cocycles of mapping classes of surfaces

TL;DR

This work extends periodic-approximation results for leading Lyapunov exponents from linear cocycles to cocycles valued in the mapping class group by exploiting a closing property for the base dynamics. It introduces a metric joint spectral radius in the MC(G) setting, realized by ergodic measures along curves in the surface, and shows Lyapunov growth can be captured by periodic orbits and by subadditive growth of Teichmüller/Thurston distances. The authors combine ergodic theory, Teichmüller geometry, and spectral theory to connect Lyapunov exponents with joint spectral-radius growth, including a thermodynamic formalism for subadditive potentials. The results suggest a broad framework for similar phenomena in non-linear group actions on geometric spaces and hint at extensions to other metrics and groups, such as Kobayashi or Hofer geometries.

Abstract

We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be approximated by periodic orbits. We also extend the notion of the joint spectral radius to this setting, interpreting it via the exponential growth of curves under iterated mapping classes. Our approach connects ideas from ergodic theory, Teichmüller geometry, and spectral theory, and suggests a broader framework for similar results.

Paper Structure

This paper contains 5 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Proofs of main results
  4. Thermodynamic formalism
  5. Potential Extensions

Key Result

Theorem 1.1

Let $v(n,g)=g_n g_{n-1} \ldots g_1$ be a product of random homeomorphisms where $g_i$ are chosen independently and distributed with a probability measure of finite first moment. Then there is (random) filtration of subsurfaces $Y_{1}\subset Y_{2}\subset...\subset Y_{k}=\Sigma$ and (deterministic) to whenever the simple closed curve $\alpha$ can be isotoped to a curve contained in $Y_{i}$ but not i

Theorems & Definitions (10)

  • Theorem 1.1: Thurston1988Karlsson-ThurstonHorbez2016
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Lemma 3.1: GG
  • proof : Proof of Theorem \ref{['main-thm']}
  • proof : Proof of Theorem \ref{['main-thm2']}
  • Corollary 3.2
  • proof