Related papers: Invariant sets for the wind-tree model

This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set $M_0$ of the billiard ball map is a topological Cantor set and its Hausdorff dimension has…

One of the versions of the wind-tree model of Boltzmann gas, suggested by Paul and Tatiana Ehrenfest more than a century ago, can be seen as a billiard in the plane endowed with $\mathbb{Z}\oplus\mathbb{Z}$-periodic rectangular obstacles.…

We consider aperiodic wind-tree models, and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions, and has a dense set of periodic points.

In recent papers, it has been shown that (i) the dynamics of theories involving gravity can be described, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space bounded by hyperplanes; and (ii) that…

In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist…

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads…

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards.…

It is known that $C^1$-smooth strictly convex Radon norms in $\mathbb{R}^2$ can be characterized by the property that the outer billiard map, which corresponds to the unit ball of the norm, has an invariant curve consisting of 4-periodic…

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace-Beltrami operator on a compact Riemannian manifold with boundary with Robin boundary conditions. Given a Kronecker invariant torus $\Lambda$ of…

In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that…

In this work we completely describe the dynamics of triangle tiling billiards. In the first part of this work, we propose a geometric approach of dynamics by introducing natural foliations associated to it. In the second part, we exploit…

The Poincar\'e problem is a model of two-dimensional internal waves in stable-stratified fluid. The chess billiard flow, a variation of a typical billiard flow, drives the formation behind and describes the evolution of these internal…

In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent. In this paper we focus on the problem of recurrence for elements of smooth curves in the moduli space. We…

This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding…

We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and…

We prove that equivalence classes of marked length isospectral Birkhoff billiard tables are compact in the $C^\infty$ topology, analogous to the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak. To do so, we derive a…

In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called…

A toy model for ferroic orders through entropy. As the rod/spin concentration (or the volume enclosing the ferroic rods) is variable by a tunable force conjugate to the order parameter, the model exhibits hysteresis associated to some…

We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of…

For Hoelder cocycles over a Lipschitz base transformation, possibly non-invertible, we show that the subbundles given by the Oseledets Theorem are Hoelder-continuous on compact sets of measure arbitrarily close to 1. The results extend to…