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Recurrence and Embeddings in Planar Wind-Tree Models

Chen Frenkel

Abstract

We study periodic infinite billiards in the plane. We show that for rational models, some particular obstacles can be added periodically, so that the billiard flow in the resulting table is recurrent in almost every direction.

Recurrence and Embeddings in Planar Wind-Tree Models

Abstract

We study periodic infinite billiards in the plane. We show that for rational models, some particular obstacles can be added periodically, so that the billiard flow in the resulting table is recurrent in almost every direction.
Paper Structure (41 sections, 16 theorems, 17 equations, 6 figures)

This paper contains 41 sections, 16 theorems, 17 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgements
  4. Background
  5. Translation surfaces and properties
  6. Constructive definition of translation surfaces
  7. Cone singularities
  8. Geometric and analytic definitions
  9. Properties of translation surfaces
  10. Translation equivalence maps and translation covering maps
  11. Affine diffeomorphism and $\mathrm{GL}(2, \mathbb{R})$-action
  12. Intersection form and orientation
  13. Holonomy map
  14. Cutting and gluing in parallel
  15. Half-translation surfaces
  16. ...and 26 more sections

Key Result

Theorem 1.1

For any rational wind-tree model $\mathrm{WT}(\Lambda, \Omega)$ one can add obstacles to $\Omega$ so that for the new configuration $\widehat{\Omega}$ the billiard flow in $\mathrm{WT}(\Lambda, \widehat{\Omega})$ is recurrent.

Figures (6)

  • Figure 1: The original wind-tree model.
  • Figure 2: $\mathrm{WT}(\Lambda, \Omega)$ (top) and $\mathrm{WT}(\Lambda, \widehat{\Omega})$ (bottom).
  • Figure 3: A $\mathbb{Z}$-cover defined by a cycle on the torus.
  • Figure 4: An example of $X$ for $\Omega$ consisting of a triangle with angles $\frac{\pi}{2},\frac{3\pi}{8},\frac{\pi}{8}$.
  • Figure 5: Example for the cover cycles.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8: KMS
  • Theorem 2.9
  • Theorem 2.10: CE
  • ...and 12 more