Related papers: A few remarks on periodic orbits for planar billia…
In this work we address the question of proving the stability of elliptic 2-periodic orbits for strictly convex billiards. Eventhough it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that…
Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is…
Optical mushroom shaped billiards offer a unique opportunity to isolate and study non-dispersive, marginally unstable periodic orbits. Here we show that the openness of the cavity to external fields presents unanticipated consequences for…
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…
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…
Recently it was proved that every billiard trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by a $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$…
We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we…
We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we…
Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant…
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…
The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic…
Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a…
In this paper, we give detailed analysis and description of periodic trajectories of the billiard system within an ellipsoid in the 3-dimensional Minkowski space, taking into account all possibilities for the caustics. The conditions for…
The billiard motion inside an ellipsoid $Q \subset \Rset^{n+1}$ is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each…
In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist…
We investigate the fundamental properties of Minkowski billiards and introduce a new coordinate system $(s,u)$ on the phase space $\mathcal{M}$. In this coordinate system, the Minkowski billiard map $\mathcal{T}$ preserves the standard area…
We propose geometric tools that are suitable for studying the behavior of a billiard trajectory in a homogeneous force field. Two examples are considered: a vertical plane with an open top and with a parabolic or right angle boundary at the…
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…
We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $\epsilon_0>0$, an $\O(\lambda^{-\epsilon_0})$ quasimode must have $L^2$ mass in the "wings" bounded below by $\lambda^{-2-\delta}$ for…
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…