Related papers: Aperiodic points for outer billiards

It is known that $C^1$-smooth strictly convex Radon norms in $\mathbb{R}^2$ can be characterized by the property that the outer billiard map, which corresponds to the unit ball of the norm, has an invariant curve consisting of 4-periodic…

In this article, we study polygonal symplectic billiards. We provide new results, some of which are inspired by numerical investigations. In particular, we present several polygons for which all orbits are periodic. We demonstrate their…

We present some foundational results about the outer length billiard system, including its generating function and the invariant area form. We describe the limiting behavior of the orbits far away from the billiard table: the orbits of the…

We discuss a recent result by C. Culter: every polygonal outer billiard has a periodic trajectory.

We consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a…

We introduce symplectic billiards for pairs of possibly non-convex polygons. After establishing basic properties, we give several criteria on pairs of polygons for the symplectic billiard map to be fully periodic, i.e. $\textit{every}$…

We prove some partial results on the periodicity of billiard systems on graphs. The results specialize to the case of $n$ billiards with equal mass on the unit interval or circle traveling at the same speed.

We compare invariants of N-periodic trajectories in the elliptic billiard, classic and new, to their aperiodic counterparts via a spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a…

We give a proof for $(2n + 1,n)$ and $(2n, n-1)$-periodic Ivrii's conjecture for planar outer billiards. We also give new simple geometric proofs for the 3 and 4-periodic cases for outer and symplectic billiards, and generalize for higher…

In this paper, we define a variant of billiards in which the ball bounces around a square grid erasing walls as it goes. We prove that there exist periodic tunnels with arbitrarily large period from any possible starting point, that there…

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…

In this paper we investigate some generic properties of a billiard system on a convex table. We show that generically, every hyperbolic periodic point admits some homoclinic orbit.

Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior…

We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer…

We introduce the iteration theory for periodic billiard trajectories in a compact and convex domain of the Euclidean space, and we apply it to establish a multiplicity result for non-iterated trajectories.

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 review some properties of periodic orbit families in polygonal billiards and discuss in particular a sum rule that they obey. In addition, we provide algorithms to determine periodic orbit families and present numerical results that shed…

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.

In the class of projective billiards, which contains the usual billiards, we exhibit counter-examples to Ivrii's conjecture, which states that in any planar billiard with smooth boundary the set of periodic orbits has zero measure. The…

A periodic orbit on a frictionless billiard table is a piecewise linear path of a billiard ball that begins and ends at the same point with the same angle of incidence. The period of a primitive periodic orbit is the number of times the…