Related papers: Non-smooth convex caustics for Birkhoff billiard

Dynamical billiards consist of a particle on a two-dimensional table, bouncing elastically off a boundary curve. The state of the system is given by two numbers: one describing the location along the curve where the bounce occurs, and…

This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals,…

We prove that a a strongly convex planar domain (Birkhoff table) with dihedral symmetry, which is sufficiently close in a finitely smooth topology to an ellipse, is deformationally spectrally rigid within the class of domains preserving…

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has…

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $K\subset \mathbb{R}^d$ has the property that the tangent cone of every non-smooth…

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…

In the class of strictly convex smooth boundaries, each of which not having strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the "normalized" Mather's $\beta$-function are invariants under…

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We…

This paper surveys our results on integrable billiards. We consider various models of billiards, including Birkhoff, outer, magnetic, and Minkowski billiards. Also, we discuss wire billiards and billiards in cones. For four models of convex…

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…

This article is concerned with the study of Mather's \beta-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it…

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not…

We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps…

We solve the longstanding problem of smoothing a stadium billiard. Besides our results demonstrate why there were no clear conjectures how much the stadium's boundary must be smoothened to destroy chaotic dynamics. To do that we needed to…

We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with…

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…

We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of…

This paper studies billiard models with a generalized law of reflection, the so-called projective billiards. They unify various laws, including the classical one in a Euclidean, pseudo-Euclidean or Riemannian metric. They were introduced…

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all…

We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrable then the boundary curve is necessarily a circle. This…