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Wind-tree tiling billiards and their trapping strips

Magali Jay

Abstract

We introduce a new dynamical system: the wind-tree tiling billiards. This system studies trajectories of a ray in Euclidean space which has a negative refractive index when encountering rectangular obstacles located at lattice points. We show that for almost every configuration of the system, trajectories with initial vertical direction are trapped in an infinite strip of the plane. This result is reminiscent of the propagation of light rays in Eaton lenses, as shown by Frączek and Schmoll.

Wind-tree tiling billiards and their trapping strips

Abstract

We introduce a new dynamical system: the wind-tree tiling billiards. This system studies trajectories of a ray in Euclidean space which has a negative refractive index when encountering rectangular obstacles located at lattice points. We show that for almost every configuration of the system, trajectories with initial vertical direction are trapped in an infinite strip of the plane. This result is reminiscent of the propagation of light rays in Eaton lenses, as shown by Frączek and Schmoll.

Paper Structure

This paper contains 25 sections, 12 theorems, 59 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Statement of the main theorem
  3. Connections to other work: Eaton lenses
  4. Strategy
  5. Background and notation
  6. Translation surfaces
  7. Zippered rectangles
  8. Teichmüller spaces and moduli spaces
  9. The Kontsevich-Zorich cocycle (geometric point of view)
  10. The Oseledets theorem
  11. Notation
  12. The corresponding surface
  13. From foliations to an appropriate slit surface
  14. Equivalent trajectories
  15. The finite half-translation surface we will work with
  16. ...and 10 more sections

Key Result

Theorem A

For almost every admissible tuple $(\Lambda,a,b,\theta)$, there exist constants ${C > 0}$ and $\Theta \in [0, \pi)$, such that every vertical trajectory in $W(\Lambda,a,b,\theta)$ is trapped in an infinite band of width $C > 0$ in direction making an angle $\Theta$ with the horizontal.

Figures (19)

  • Figure 1: Setting of our systems
  • Figure 2: A few steps of tiling billiards trajectory
  • Figure 3: Example of a trajectory, trapped in a strip
  • Figure 4: A dynamical system with Eaton lenses
  • Figure 5: Comparison of the refraction behavior of our rectangle with an Eaton lens
  • ...and 14 more figures

Theorems & Definitions (30)

  • Theorem A
  • Theorem B
  • Definition 2.1: IET
  • Theorem 2.2: Oseledets
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 20 more