相关论文: On Estimates of the Number of Collisions for Billi…

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.

This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is natural to search for bodies that are, in a sense, close to invisible. We…

A numerically efficient Fredholm formulation of the billiard problem is presented. The standard solution in the framework of the boundary integral method in terms of a search for roots of a secular determinant is reviewed first. We next…

Astute variations in the geometry of mathematical billiard tables have been and continue to be a source of understanding their wide range of dynamical behaviors, from regular to chaotic. Viewing standard specular billiards in the broader…

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…

Polygonalization of any smooth billiard boundary can be carried out in several ways. We show here that the semiclassical description depends on the polygonalization process and the results can be inequivalent. We also establish that…

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…

This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary,…

We derive semiclassical contributions of periodic orbits from a boundary integral equation for three-dimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index…

We give an optical physicist view of the problem of the trajectories in a polygonal billiard using only basic facts of Optics and the theory of functions of a complex variable. This approach allow us to stablish a certain correspondence…

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth…

We introduce a new class of billiard-like system, ``bouncing outer billiards" which are 3-dimensional cousins of outer billiards of Neumann and Moser. We prove that bouncing outer billiard on a smooth convex body has at least four…

We provide explicit lower estimates on the complexity growth in typical directions for a class of irrational triangle billiards

We consider outer billiard outside regular convex polygons. We deal with the case of regular polygons with $\{3,4,5,6,10\}$ sides, and we describe the symbolic dynamics of the map and compute the complexity of the language.

In this text we study billiards on ovals and investigate some consequences of a rotational symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits…

We study billiards on polytopes in $\Rr^d$ with contracting reflection laws, i.e. non-standard reflection laws that contract the reflection angle towards the normal. We prove that billiards on generic polytopes are uniformly hyperbolic…

We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. We give new results on periodicity and boundedness of orbits which suggest…

In this article we study the dynamics of one-dimensional relativistic billiards containing particles with positive and negative energy. We study configurations with two identical positive masses and symmetric positions with two massless…

We consider a billiard model of a self-bound, interacting three-body system in two spatial dimensions. Numerical studies show that the classical dynamics is chaotic. The corresponding quantum system displays spectral fluctuations that…

We study the Lawn Mowing Problem restricted to periodic billiard paths in the unit square. Given the combinatorial data of a trajectory, we determine the optimal covering radius, and identify the shortest path that covers the square for any…