Divergent directions in some periodic wind-tree models

TL;DR

This paper investigates the wind-tree billiard model, showing that for certain parameters, there exist sets of angles with positive Hausdorff dimension where trajectories are self-avoiding and divergent, contrasting with recurrence in almost every direction.

Contribution

It demonstrates the existence of sets of angles with positive Hausdorff dimension leading to divergent, self-avoiding trajectories in the wind-tree model for specific parameters.

Findings

Existence of sets of angles with positive Hausdorff dimension where trajectories are self-avoiding.

For many parameters, trajectories in these sets are divergent.

Contrasts with previous results showing recurrence for almost every direction.

Abstract

The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy (arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard flow in T(a,b) is recurrent in almost every direction. We prove that for many parameters (a,b) there exists a set S of angles of positive Hausdorff dimension such that every billiard trajectory in T(a,b) with initial angle in S is self-avoiding. In particular, the flow in a direction of S is divergent.