Diffusion rate in non-generic directions in the wind-tree model

TL;DR

This paper demonstrates that any real number in [0,1) can be a diffusion rate in the wind-tree model with rational parameters and provides a criterion for the Lyapunov spectrum shape, revealing full spectrum interior in certain billiards.

Contribution

It introduces a criterion for the Lyapunov spectrum of cocycles and explicitly describes the spectrum's interior for specific wind-tree billiards, a novel achievement in two-dimensional dynamics.

Findings

Any real number in [0,1) is a diffusion rate for the wind-tree model with rational parameters.

The shape of the Lyapunov spectrum can be explicitly described using the proposed criterion.

The interior of the Lyapunov spectrum can be the full square (0,1)^2 for certain billiards.

Abstract

We show that any real number in [0,1) is a diffusion rate for the wind-tree model with rational parameters. We will also provide a criterion in order to describe the shape of the Lyapunov spectrum of cocycles obtained as suspension of a representation. As an application, we exhibit an infinite family of wind-tree billiards for which the interior of the Lyapunov spectrum is a big as possible: this is the full square (0,1)^2. To the best of the knowledge of the authors, these are the first complete descriptions where the interior of the Lyapunov spectrum is known explicitly in dimension two, even for general Fuchsian groups.