Related papers: Billiards in a circle with trajectories circumscri…

We show ergodicity of (asymmetric) lemon billiards, billiard tables that are the intersection of two circles of which one contains the centers of both. These do not satisfy the Wojtkowski criteria for hyperbolicity, but we establish…

We illustrate the theory of one-dimensional pluri-Lagrangian systems with the example of commuting billiard maps in confocal quadrics.

We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes…

Physicists have used billiards to understand and explore both classical and quantum chaos. Recently, in 2001, a group at the University of Texas introduced an experimental set up for modeling the wedge billiard geometry called optical…

We examine the possible trajectories of a classical particle, trapped in a two-dimensional infinite rectangular well, using the Hamilton-Jacobi equation. We observe that three types of trajectories are possible: periodic orbits, open orbits…

We obtain an upper bound of the number of collisions of any billiard trajectory in a polyhedral angle in terms of the minimal eigenvalue of a positive definite matrix which characterizes the angle. Elements of the matrix are scalar products…

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are…

We consider the dual billiard map with respect to a smooth strictly convex closed hypersurface in linear 2m-dimensional symplectic space and prove that it has at least 2m distinct 3-periodic orbits.

Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. When the shape is a polygon, the system has a combinatorial flavor. In the polygonal case, there is a natural acceleration of the map, a…

I apply the algebraic framework developed in [1] to study geometry of hyperbolic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is described in…

In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in…

For every quadrilateral sufficiently close to a rectangle, we shall show that it possess a periodic billiard path. This is an REU work done at ICERM in Summer 2012.

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange…

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols…

Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a…

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflection from a boundary. For billiards in non-convex areas bounded by segments of confocal quadrics are studied. The topology…

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an…

Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective…

We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…

It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than…