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 H hyp (2g − 2) or 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 Ehrenfest wind-tree model, a billiard in the plane endowed with Z 2 -periodically located identical rectangular obstacles.