Perfect Retroreflectors and Billiard Dynamics

TL;DR

This paper constructs special billiard domains that almost always reverse particle directions, with exit velocities opposite to entry and near-original exit points, using asymptotic analysis of irrational circle rotations.

Contribution

It introduces a new class of semi-infinite billiard domains that achieve near-perfect retroreflection through asymptotic analysis of entrance times.

Findings

Most particles leave after finite reflections

Exit velocity is opposite to entrance velocity with high probability

Exit points are arbitrarily close to initial points

Abstract

We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in a limit when the number of iterates tends to infinity and the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.