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

## TL;DR

This paper demonstrates a non-varying phenomenon in counting cylinders passing through specific points on hyperelliptic translation surfaces and applies it to periodic trajectories in the wind-tree billiard model.

## Contribution

It establishes a non-varying phenomenon for cylinder counts passing through Weierstrass points and applies this to the wind-tree billiard model, linking geometric and dynamical properties.

## Key findings

- Non-varying phenomenon for cylinder counts on hyperelliptic surfaces
- Application to periodic trajectories in the wind-tree billiard model
- Explicit formulas for weighted counts in specific moduli spaces

## 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.