Related papers: Billiards and Hofer's Geometry
E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be…
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…
We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the…
We study the geometry of reflection of a massive point-like particle at conic section boundaries. Thereby the particle is subjected to a central force associated with either a Kepler or Hooke potential. The conic section is assumed to have…
In this paper we study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with three, mutually intersecting and mutually orthogonal cylindrical scatterers removed from it. The…
The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko's school, in particular of V.…
Let $K\subset\mathbb R^n_q$, $T\subset\mathbb R^n_p$ be two bounded strictly convex bodies (open subsets) with $C^6$-smooth boundaries. We consider the product $\overline K\times\overline T\subset\mathbb R^{2n}_{q,p}$ equipped with the…
We introduce an algebraic formulation of billiards on plane curves over algebraically closed fields, extending Glutsyuk's complex billiards. For any smooth algebraic curve $C$ of degree $d \geq 2$, algebraic billiards is a rational…
We provide an upper bound on the Lagrangian Hofer distance between equators in the cylinder in terms of the barcode of persistent Floer homology. The bound consists of a weighted sum of the lengths of the finite bars and the spectral…
We consider a remarkable $C^2$-smooth billiard table introduced by Hans L.Fetter. It is obtained by the string construction from a regular hexagon for a special value of the length of the string. It was suggested as a possible…
We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension.
We rigorously investigate closed Minkowski/Finsler billiard trajectories on $n$-dimensional convex bodies. We outline the central properties in comparison and differentiation from the Euclidean special case and establish two main results…
In this paper we study the area function of polygons, where the vertices are sliding along curves. We give geometric criteria for the critical points and determine also the Hesse matrix at those points. This is the starting point for a…
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a…
We study billiards in domains enclosed by circular polygons. These are closed $C^1$ strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories…
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of…
We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we…
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…
A periodic trajectory on a polygonal billiard table is stable if it persists under any sufficiently small perturbation of the table. It is a standard result that a periodic trajectory on an $n$-gon gives rise in a natural way to a closed…
We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled…