Related papers: Dual billiards, twist maps, and impact oscillators

We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with…

We develop a theory of T-duality for transitive Courant algebroids. We show that T-duality between transitive Courant algebroids E\rightarrow M and \tilde{E}\rightarrow \tilde{M} induces a map between the spaces of sections of the…

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 the present paper we introduce a new generating function for outer billiards in the plane. Using this generating function, we prove the following rigidity result: if the vicinity of the smooth convex plane curve $\gamma$ of positive…

We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties…

Single and double-slit experiments are performed with two microwave billiards with the shapes of a rectangle, respectively, a quarter stadium. The classical dynamics of the former is regular, that of the latter is chaotic. Microwaves can…

A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A…

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.

We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted…

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…

Recently the duality map between electric-like asymptotic charges of $p$-form gauge theories is studied. The outcome is an existence and uniqueness theorem and the topological nature of the duality map. The goal of this work is to extend…

We compute accurately the golden critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for…

We study billiards in plane domains, with a perpendicular magnetic field and a potential. We give some results on periodic orbits, KAM tori and adiabatic invariants. We also prove the existence of bound states in a related scattering…

The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear,…

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result…

We determine the rings of invariants in the symmetric algebra on the dual of a vector space V over the field of two elements, for the group G of orthogonal transformations preserving a non-singular quadratic form on V. The invariant ring is…

The geometry of a billiard boundary fundamentally governs its dynamics, ranging from integrable to mixed and fully chaotic regimes. Bean- and peanut-shaped billiards have varying curvature with both focusing and defocusing walls without a…

The well-known twenty types of 2-uniform tilings of the plane give rise infinitely many doubly semi-equivelar maps on the torus. In this article, we show that every such doubly semi-equivelar map on the torus contains a Hamiltonian cycle.…

Compactifications in duality covariant constructions such as generalised geometry and double field theory have proven to be suitable frameworks to reproduce gauged supergravities containing non-geometric fluxes. However, it is a priori…

We consider a strictly convex billiard table with $C^2$ boundary, with the dynamics subjected to random perturbations. Each time the billiard ball hits the boundary its reflection angle has a random perturbation. The perturbation…