Related papers: On complex algebraic caustics in planar and projec…
Billiards in ellipses have a confocal ellipse or hyperbola as caustic. The goal of this paper is to prove that for each billiard of one type there exists an isometric counterpart of the other type. Isometry means here that the lengths of…
In this paper, we continue to study billiards inside cones $K\subset \mathbb{R}^n$ over strictly convex closed $C^3$ manifolds with non-degenerate second fundamental form. Recently we proved that the billiard is superintegrable, i.e., the…
We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of…
By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a…
Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant…
We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential…
Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from…
In this paper I will unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then…
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…
A variety of mesoscopic systems can be represented as a billiard with a random coupling to the exterior at the boundary. Examples include quantum dots with multiple leads, quantum corrals with different kinds of atoms forming the boundary,…
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…
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and…
The connected configuration space of a so called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with…
We consider the integrable dynamics of a Kepler billiard in the plane bounded by a branch of a conic section focused at the Kepler center. We show that in this case, for non-zero-energy orbits, the lines of consecutive second orbital foci…
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…
We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special…
The most general solution to the Einstein equations in $4=3+1$ dimensions in the asymptotical limit close to the cosmological singularity under the BKL (Belinski-Khalatnikov-Lifshitz) hypothesis, for which space gradients are neglected and…
Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years polygonal billiards have attracted great attention due to their application in the understanding of anomalous transport, but also at the fundamental level,…
We design a computational experiment in which a quantum particle tunnels into a billiard of variable shape and scatters out of it through a double-slit opening on the billiard's base. The interference patterns produced by the scattered…
Ivrii's Conjecture states that in every billiard in Euclidean space the set of periodic orbits has measure zero. It implies that for every $k\geq2$ there are no k-reflective billiards, i.e., billiards having an open set of k-periodic…