Related papers: Recurrence and Embeddings in Planar Wind-Tree Mode…

We give a new proof for the directional billiard complexity in the cube, which was conjectured in \cite{Ra} and proven in \cite{Ar.Ma.Sh.Ta}. Our technique gives us a similar theorem for some rational polyhedra.

We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming…

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…

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,…

In this paper we show that all infinite trees which have bounded coordination and whose surface is negligible with respect to the volume in the limit of large distances (so that they can be embedded in a finite-dimensional euclidean space)…

We study the billiard dynamics in annular tables between two excentric circles. As the center and the radius of the inner circle change, a two parameters map is defined by the first return of trajectories to the obstacle. We obtain an…

We define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality…

We introduce a new dynamical system that we call "tiling billiards," where trajectories refract through planar tilings. This system is motivated by a recent discovery of physical substances with negative indices of refraction. We…

We use Ratner's theorem to compute the asymptotics of the number of (cylinders of) periodic trajectories in a rectangle with a barrier, assuming that the location p/q of the barrier is rational. We also show that as q tends to infinity, the…

We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with…

This paper is the third in a series which explores a combinatorial method for generating lattice polygons in the plane. I call this method the plaid model. In this paper I prove the main result I had been aiming for since the beginning,…

We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature. For instance, we obtain power bounds of degree two plus…

We show that the typical wind-tree model, in the sense of Baire, is recurrent and has a dense set of periodic orbits. The recurrence result also holds for the Lorentz gas : the typical Lorentz gas, in the sense of Baire, is recurrent. These…

We consider a slowly rotating rectangular billiard with moving boundaries and use the canonical perturbation theory to describe the dynamics of a billiard particle. In the process of slow evolution certain resonance conditions can be…

We investigate a rotated, orthogonal gravitational wedge billiard - a special case of the asymmetric wedge billiard - in which the dynamics are integrable. We derive equations and conditions under which periodic orbits may be constructed…

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They…

We study a particle moving at unit speed in a self-similar Lorentz billiard channel; the latter consists of an infinite sequence of cells which are identical in shape but growing exponentially in size, from left to right. We present…

We propose geometric tools that are suitable for studying the behavior of a billiard trajectory in a homogeneous force field. Two examples are considered: a vertical plane with an open top and with a parabolic or right angle boundary at the…

Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian…

In this text we study billiards on ovals and investigate some consequences of a rotational symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits…