Related papers: Back to Boundaries in Billiards

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the…

This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some basic properties…

We show that for a rational polygonal billiard, the set of pairs of points that do not illuminate each other (not connected by a billiard trajectory) is finite, and use the same method to extend the results of Leli\`evre, Monteil and Weiss,…

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and…

We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of…

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic -- enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are…

In this work, we construct linearly stable periodic orbits in $3$-dimensional domains with boundaries containing focusing components (small pieces of a sphere) where we place these components arbitrarily far apart. It demonstrates that we…

We study a class of elliptic billiards with a Keplerian potential inside, considering two cases: a reflective one, where the particle reflects elastically on the boundary, and a refractive one, where the particle can cross the billiard's…

We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times…

In this paper we study the singularity manifolds of multidimensional strictly dispersing billiards and show that the proof of the Fundamental theorem for dispersing billiards remain valid for a dense set of finitely smooth scatterers.

We investigate the effect of white-noise perturbations on chaotic trajectories in open billiards. We focus on the temporal decay of the survival probability for generic mixed-phase-space billiards. The survival probability has a total of…

We study billiards in domains enclosed by circular polygons. These are closed $C^1$ strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories…

The paper establishes the property of splittability of billiard boundary sequences in n dimensional cube into subsequences of fractional parts. This reveals a new property of integrable and weak perturbated Hamilton systems: under a simple…

We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties…

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…

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,…

Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain…

We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps…

We investigate the rotation sets of open billiards in $\mathbb{R}^N$ for the natural observable related to a starting point of a given billiard trajectory. We prove that the general rotation set is convex and the set of all convex…

We investigate statistical properties of several classes of periodic billiard models which are diffusive. An introductory chapter gives motivation, and then a review of statistical properties of dynamical systems is given in chapter 2. In…