A non-varying phenomenon with an application to the wind-tree model

Abstract

We exhibit a non-varying phenomenon for the counting problem of cylinders, weighted by their area, passing through two marked (regular) Weierstrass points of a translation surface in a hyperelliptic connected component ${\mathcal{H}}^{hyp}(2g-2)$ or ${\mathcal{H}}^{hyp}(g-1,g-1)$, $g>1$. As an application, we obtain the non-varying phenomenon for the counting problem of (weighted) periodic trajectories on the classical wind-tree model, a billiard in the plane endowed with ${\mathbb{Z}}^2$-periodically located identical rectangular obstacles.

14 pages, 3 figure. Mayor revision: Sections 3 and 4 from former version are treated uniformly on Section 3 of the new version, by working directly on the sphere (referee observation). Minor mistake on the counterexamples section corrected and new examples provided