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Shrinking targets on square-tiled surfaces

Josh Southerland

Abstract

We study a shrinking target problem on square-tiled surfaces. We show that the action of a subgroup of the Veech group of a regular square-tiled surface exhibits Diophantine properties. This generalizes the work of Finkelshtein, who studied a similar problem on the flat torus.

Shrinking targets on square-tiled surfaces

Abstract

We study a shrinking target problem on square-tiled surfaces. We show that the action of a subgroup of the Veech group of a regular square-tiled surface exhibits Diophantine properties. This generalizes the work of Finkelshtein, who studied a similar problem on the flat torus.

Paper Structure

This paper contains 14 sections, 18 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definition of a translation surface
  3. The $SL_2(\mathbb{R})$-action
  4. The Dynamical System
  5. Action of the Veech Group
  6. Acknowledgements
  7. Shrinking Targets
  8. Set-up
  9. A Brief History
  10. Application to Square-tiled Surfaces
  11. Properties of the Veech group
  12. Spectral estimates of the Koopman representation
  13. A covering argument
  14. Borel-Cantelli argument

Key Result

Theorem 1.1

Let $(X,\omega)$ be a regular square-tiled surface, and let $\Gamma$ be a subgroup of the Veech group $SL(X,\omega)$ with critical exponent $\delta_{\Gamma} > 0$. For any $y \in X$, for Lebesgue a.e. $x \in X$, the set is where $\lvert \lvert \, \cdot \, \rvert \rvert$ is the operator norm of $g$ (as a linear transformation on $\mathbb{R}^2$).

Figures (5)

  • Figure 1: Translation surface
  • Figure 2: Stabilizing element of the Veech group
  • Figure 3: Hitting the target
  • Figure 4: Hitting the target
  • Figure 5: Will hit the target (Application of $T^{-1}$ to target)

Theorems & Definitions (30)

  • Definition 1.1: Square-tiled Surface Ma18, Zm11
  • Definition 1.2: Square-tiled Surface Ma18, Zm11
  • Definition 1.3: Critical Exponent, $\delta_{\Gamma}$
  • Theorem 1.1
  • Remark 1.1
  • Theorem 2.1: Poincaré recurrence
  • Lemma 2.1: Borel-Cantelli lemma and partial converse
  • Theorem 2.2: Quantitative Borel-Cantelli lemma
  • Definition 2.1: Borel-Cantelli At09, Fa06
  • Theorem 3.1
  • ...and 20 more