Invariant sets for the wind-tree model

Abstract

We consider the wind-tree model, a ${\mathbb{Z}}^2$ - periodic billiard. In the case when the underlying compact translation surface lies on a periodic orbit of the Teichmüller geodesic flow, and at least one of the two homology classes defining the ${\mathbb{Z}}^2$ - cover is unstable for the Kontsevich-Zorich cocycle, we prove that every orbit closure of the billiard has Hausdorff dimension strictly smaller than 2. The proof relies on a construction of explicit invariant functions, which along the way gives a new proof of non-ergodicity and non-transitivity of the wind-tree model for all parameters and almost all directions, as first shown by Frcaczek and Ulcigrai (2014).

20 pages, 4 figues; comments welcome!