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.