Related papers: Outer billiard outside regular polygons

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 study outer length billiards; our main results are as follows. We prove 3- and 4-periodic versions of the Ivrii conjecture. We show that, for every period $n\ge 3$, there exists a functional space of billiard tables that possess…

We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same…

Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The…

We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the…

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…

In this paper we present new results regarding the periodicity of outer billiards in the hyperbolic plane around polygonal tables which are tiles in regular two-piece tilings of the hyperbolic plane.

We focus on the outer length billiard dynamics, acting on the exterior of a strictly-convex planar domain. We first show that ellipses are totally integrable. We then provide an explicit representation of first order terms for the formal…

Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has…

This paper explores the number of parallelograms that appear in a billiard path that enters one corner of a rectangle and leaves a second corner of a rectangle as a function of the normalized dimensions of the rectangle.

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…

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic,…

We study polygonal billiards with one-sided vertical mirror scattered on a square billiard table. We associate trajectories of these kinds of billiards with double rotations and study orbit behavior and questions of complexity.

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…

We consider a non-minimal billiard trajectory inside the cube. We study the language of the associated orbit when the map is coded with three letters associated to three non-parallel faces of the cube.

We consider the billiard map in the hypercube of $\mathbb{R}^d$. We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3d-3}$ is the order…

It is shown that the set of 4-period orbits in outer billiard with piecewise smooth convex boundary has an empty interior, provided that no four corners of the boundary form a parallelogram.

The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given…

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…

We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in…