The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion

TL;DR

This paper investigates the dynamical behavior of periodic wind-tree billiard models, demonstrating the existence of recurrent and escaping trajectories depending on parameters, and extending results to a dense set of configurations.

Contribution

It provides new results on periodic directions, recurrence, and escape rates in wind-tree models for various rational parameters, expanding understanding of their dynamical complexity.

Findings

Existence of completely periodic directions for certain rational parameters.

Recurrence of trajectories in specific parameter classes.

Escape rates for almost all directions in some parameter regimes.

Abstract

We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely periodic directions, and recurrence; for another class of rational parameters, there are directions in which all trajectories escape, and we prove a rate of escape for almost all directions. These results extend to a dense $G_\delta$ of parameters.