Invariant sets for the wind-tree model

Yuriy Tumarkin

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arXiv · 2510.10760

Invariant sets for the wind-tree model

Yuriy Tumarkin

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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 Teichm\"uller 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 Fr\c{a}czek and Ulcigrai (2014).

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Yuriy Tumarkin. 2025-10-12. Invariant sets for the wind-tree model. https://arxiv.org/abs/2510.10760

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