Related papers: Outer Billiards on the Penrose Kite: Compactificat…
We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic…
We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if,…
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…
We give a simple proof of our previous result with V. Zharnitsky that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a…
We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all…
Motivated by open problems in F-theory, we reconsider warped compactifications of M theory on 8-manifolds to AdS3 spaces in the presence of a non-trivial field strength of the M-theory 3-form, studying the most general conditions under…
We show that for almost every $(P,\lambda)$ where $P$ is a convex polygon and $\lambda\in(0,1)$, the corresponding outer billiard about $P$ with contraction $\lambda$ is asymptotically periodic, i.e., has a finite number of periodic orbits…
In this short paper, we show a characterization of Birkhoff Billiards inside discs which is related to the expansion of the formal Lazutkin conjugacy at the boundary.
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.
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.
An existence of an aperiodic point for outer billiard outside regular dodecagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic, and all possible periods are listed explicitly. The proof is…
This paper addresses the question of genericity of existence of elliptic islands for the billiard map associated to strictly convex closed curves. More precisely, we study 2-periodic orbits of billiards associated to C5 closed and strictly…
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…
Orbits in different dispersive billiard systems, e.g. the 3 disk system, are mapped into a topological well ordered symbol plane and it is showed that forbidden and allowed orbits are separated by a monotone pruning front. The pruning front…
We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The…
The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.
The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a…
We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross…
We study outer billiard systems around a class of circular sectors. For semi-discs, we prove the existence of elliptic islands occupying a positive proportion of the plane. Combined with known results, this shows the coexistence of…
Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even…