Related papers: Invariant sets for the wind-tree model

We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with…

We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic…

The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy…

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed…

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes…

In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism,…

We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely periodic directions, and recurrence; for another class of…

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…

In this paper, we consider generalized wind-tree models and $\Z^d$-covers over compact translation surfaces. Under suitable hypothesis, we prove recurrence of the linear flow in a generic direction and non-ergodicity of Lebesgue measure.

There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = {\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix}}$. We classify the U-invariant ergodic measures on certain…

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory" (from 4…

We study periodic infinite billiards in the plane. We show that for rational models, some particular obstacles can be added periodically, so that the billiard flow in the resulting table is recurrent in almost every direction.

The set of all possible configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology is a compact metric space. For a typical configuration we show that the wind-tree dynamics has infinite ergodic index in almost…

We show that the Hausdorff dimension of any proper Teichm\"uller horocycle flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the…

In the Ehrenfest wind tree model, a point particle moves on the plane and collides with randomly placed fixed square obstacles under the usual law of geometric optics. The particle represents the wind and the squares are the trees. We…

The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers disposed at each integer point. We prove that independently of the size of the scatterers, generically with respect to the angle, the…

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not…

In this paper we prove that in any analytic one-parameter family of twist maps of the annulus, homotopically invariant curves filled with periodic points corresponding to a given rotation number, either exist for all values of the…

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially…

We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if,…