Related papers: Periodic Orbits of Billiards on an Equilateral Tri…
We apply a recently developed semiclassical theory of short peridic orbits to the stadium billiard. We give explicit expresions for the resonances of periodic orbits and for the application of the semiclassical Hamiltonian operator to them.…
Periodic orbits are the central ingredients of modern semiclassical theories and corrections to these are generally non-classical in origin. We show here that for the class of generic polygonal billiards, the corrections are predominantly…
A simplex is the convex hull of $n+1$ points in $\mathbb{R}^{n}$ which form an affine basis. A regular simplex $\Delta^n$ is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of $\mathbb{R}^n$.…
A family of periodic orbits is proven to exist in the spatial lunar problem that are continuations of a family of consecutive collision orbits, perpendicular to the primary orbit plane. This family emanates from all but two energy values.…
We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and…
We show that polynomial recursions $x_{n+1}=x_{n}^{m}-k$ where $k,m$ are integers and $m$ is positive have no nontrivial periodic integral orbits for $m\geq3$. If $m=2$ then the recursion has integral two-cycles for infinitely many values…
Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20…
Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery…
We give an explicit sub-exponential estimate on the growth rate of periodic orbits and generalized diagonals for typical triangle billiards.
We consider a cubic nonlinear wave equation on a network and show that inspecting the normal modes of the graph, we can immediately identify which ones extend into nonlinear periodic orbits. Two main classes of nonlinear periodic orbits…
Since Poincar\'e, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at…
One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg…
It has been known since Fagnano in 1775 that an acute triangle always has a periodic billiard path, namely the orthic triangle. It is currently unknown whether every obtuse triangle has a periodic path. In 2006, Schwartz showed that every…
Euclidean outer billiard on a regular polygon (that is not a triangle, square or a hexagon) has aperiodic points, i.e., points where all iterates of the outer billiard map are defined and yield pairwise distinct images. This result answers…
In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the…
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard…
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,…
We give an optical physicist view of the problem of the trajectories in a polygonal billiard using only basic facts of Optics and the theory of functions of a complex variable. This approach allow us to stablish a certain correspondence…
We investigate a combinatorial optimization problem that involves patrolling the edges of an acute triangle using a unit-speed agent. The goal is to minimize the maximum (1-gap) idle time of any edge, which is defined as the time gap…
The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex algebraic version of…