Invariant sets for the wind-tree model

2025/10/12 by Tumarkin, Yuriy
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.10760

Abstract

We consider the wind-tree model, a ℤ² - periodic billiard. In the case when the underlying compact translation surface lies on a periodic orbit of the Teichmüller geodesic flow, and at least one of the two homology classes defining the ℤ² - 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 Frcaczek and Ulcigrai (2014).

Citations

• Diffusion rate in non-generic directions in the wind-tree model
• Recurrence and non-ergodicity in generalized wind-tree models
• Isolation, equidistribution, and orbit closures for the SL(2,R) action\n on Moduli space
• Every flat surface is Birkhoff and Oseledets generic in almost every\n direction
• Non-ergodic Z-periodic billiards and infinite translation surfaces
• Linearization of generalized interval exchange maps
• Generalized staircases: recurrence and symmetry
• The cohomological equation for Roth type interval exchange maps

Related

• Non-ergodic Z-periodic billiards and infinite translation surfaces
• Recurrence and non-ergodicity in generalized wind-tree models
• Recurrence and Embeddings in Planar Wind-Tree Models
• Diffusion rate in non-generic directions in the wind-tree model
• Infinite ergodic index of the ehrenfest wind-tree model