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On Convergence of Random Walks on Moduli Space

Roland Prohaska

Abstract

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the $n$-step distributions from almost every starting point in an affine invariant submanifold towards the associated affine invariant measure, and almost sure pathwise equidistribution towards the affine invariant measure on the $SL_2(\mathbb{R})$-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.

On Convergence of Random Walks on Moduli Space

Abstract

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the -step distributions from almost every starting point in an affine invariant submanifold towards the associated affine invariant measure, and almost sure pathwise equidistribution towards the affine invariant measure on the -orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.

Paper Structure

This paper contains 3 sections, 10 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Generic Points
  3. Pathwise Equidistribution

Key Result

Theorem \oldthetheorem

Let $\mu$ be an absolutely continuous compactly supported $\mathop{\mathrm{SO}}\nolimits_2(\mathbb{R})$-bi-invariant probability measure on $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{R})$ and $x\in\mathcal{H}$. Then the orbit closure $\overline{\mathop{\mathrm{SL}}\nolimits_2(\mathbb{R})x}$ is an affi as $n\to\infty$.

Theorems & Definitions (18)

  • Theorem \oldthetheorem: Eskin--Mirzakhani--Mohammadi EMM
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof : Proof of Proposition \ref{['prop:spectral_gap']}
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: EMi
  • Corollary \oldthetheorem
  • proof
  • ...and 8 more