Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant sets for the wind-tree model

Yuriy Tumarkin

TL;DR

This work studies the wind-tree model by folding the billiard into a translation-surface framework and analyzing it along periodic Teichmüller orbits. It constructs explicit continuous invariant functions on suitable $ ext{Z}^d$-covers using stable components of the Kontsevich-Zorich cocycle and coboundary theory, then leverages a Vershik-adic coding and a Cantor-set argument to bound the Hausdorff dimension of orbit closures to below 2 in the periodic/unstable setting. The results provide a new, constructive proof of non-ergodicity and non-transitivity for all parameters and almost all directions, and extend the analysis to broader covers via Roth-type coboundaries and block-splitting of homology. The approach combines Rauzy-Veech dynamics, cohomological transfer functions, and explicit invariant-set plotting to yield quantitative geometric insight into wind-tree dynamics and its invariant sets.

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 Teichmüller 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 Frcaczek and Ulcigrai (2014).

Invariant sets for the wind-tree model

TL;DR

This work studies the wind-tree model by folding the billiard into a translation-surface framework and analyzing it along periodic Teichmüller orbits. It constructs explicit continuous invariant functions on suitable -covers using stable components of the Kontsevich-Zorich cocycle and coboundary theory, then leverages a Vershik-adic coding and a Cantor-set argument to bound the Hausdorff dimension of orbit closures to below 2 in the periodic/unstable setting. The results provide a new, constructive proof of non-ergodicity and non-transitivity for all parameters and almost all directions, and extend the analysis to broader covers via Roth-type coboundaries and block-splitting of homology. The approach combines Rauzy-Veech dynamics, cohomological transfer functions, and explicit invariant-set plotting to yield quantitative geometric insight into wind-tree dynamics and its invariant sets.

Abstract

We consider the wind-tree model, a - 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 - 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).

Paper Structure

This paper contains 20 sections, 19 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background on translation surfaces and IETs
  4. Teichmüller flow
  5. Teichmüller space and moduli space
  6. The $SL(2,\mathbb{R})$ action
  7. The Kontsevich-Zorich cocycle
  8. Orbit closures and $SL(2,\mathbb{R})$-invariant subbundles
  9. Generic properties of a translation surface
  10. IETs
  11. Stable space and coboundaries
  12. Covers
  13. The wind-tree model
  14. Construction of invariant sets and non-transitivity
  15. Hausdorff dimension in the periodic case
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $a,b \in (0,1)$ be any parameters, let $\theta$ be a direction such that $r_{\pi/2-\theta} X(a,b)$ is periodic for the Teichmüller geodesic flow and at least one of $\gamma_h$ and $\gamma_v$ is unstable. Then for the billiard flow on the wind-tree model with parameters $a,b$ in direction $\theta

Figures (4)

  • Figure 1: A $400\times 400$ window of an invariant set for the billiard in the wind-tree model, in the $g_t$-periodic case.
  • Figure 2: An example of towers for an IET on 3 intervals. The union of the base intervals is $I^{(k)}$, and the union of all the floors (including the base) is $I^{(r)}$. From the towers we can not see exactly the image under $T^{(r)}$ of the top floor of a tower, but we do know that it gets sent into $I^{(k)}$, hence somewhere in the union of the bases.
  • Figure 3: The surface $Y$ with the gluings and curves $v,h,c_h$ and $c_v$.
  • Figure 4: The surface $X$ with its four singularities and the classes generating $H_1(X,\mathbb{Z})$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Theorems 1.1 and 1.2 in ChaikaEskin
  • Theorem 2.5: Theorem 2 in KMS
  • Lemma 2.6
  • Theorem 2.7: See Theorem A in MMY1 and Corollary 3.6 in MMY2
  • Theorem 2.8
  • ...and 24 more