Wind-tree tiling billiards and their trapping strips

TL;DR

This paper introduces wind-tree tiling billiards, a new dynamical system modeling ray trajectories with negative refraction, demonstrating that most initial directions lead to trapping in infinite strips, akin to light propagation in Eaton lenses.

Contribution

The paper presents a novel dynamical system and proves that for almost all configurations, trajectories with vertical initial directions are trapped, expanding understanding of billiard dynamics with negative refraction.

Findings

Trajectories with vertical initial directions are trapped in infinite strips.

The system models negative refraction in a billiard setting.

Results relate to light propagation in Eaton lenses.

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 Fr\k{a}czek and Schmoll.