Related papers: Periodic Billiards in Isosceles Triangles
We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not…
This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace…
We explore the triangle outer billiards map in points at infinity in the hyperbolic plane, focusing on the rotation number. Building on Dogru and Tabachnikov's work, which established the conditions for triangles where the rotation number…
For a triangle $\Delta$, let (P) denote the problem of the existence of points in the plane of $\Delta$, that are at rational distance to the 3 vertices of $\Delta$. Answer to (P) is known to be positive in the following situation: $\Delta$…
Periodic classical trajectories are of fundamental importance both in classical and quantum physics. Here we develop path integral techniques to investigate such trajectories in an arbitrary, not necessarily energy conserving hamiltonian…
A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing…
We study billiards in plane domains, with a perpendicular magnetic field and a potential. We give some results on periodic orbits, KAM tori and adiabatic invariants. We also prove the existence of bound states in a related scattering…
In the present paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major…
The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric,…
We study generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit.
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…
The structure of the semiclassical trace formula can be used to construct a quasi-classical evolution operator whose spectrum has a one-to-one correspondence with the semiclassical quantum spectrum. We illustrate this for marginally…
We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular…
In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp for closed geodesics on negatively curved compact surfaces. The first of these estimates…
The existence and bifurcation of homoclinic orbits in planar piecewise linear homogeneous systems with two regions separated by a discontinuity boundary are investigated in this paper. In addition, existence of periodic orbits and stability…
As a contribution to the inverse scattering problem for classical chaotic systems, we show that one can select sequences of intervals of continuity, each of which yields the information about period, eigenvalue and symmetry of one unstable…
Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in…
Revised version: some minor errors and typos fixed; exposition watered. Abstract: To a trajectory of a billiard in parallelogram we assign its symbolic trajectory - the sequence of numbers of coordinate plane, to which the faces met by the…
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…
Several examples of pairs of isospectral planar domains have been produced in the two-dimensional Euclidean space by various methods. We show that all these examples rely on the symmetry between points and blocks in finite projective…