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All the segments (or their continuations) of a billiard trajectory inside an ellipsoid of $\Rset^n$ are tangent to n-1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated to periodic billiard…
In this paper the problem of estimating the number of periodical billiard trajectories is considered. The main result is the theorem on Morse theory for periodical billiard trajectories.
The semiclassical description of billiard spectra is extended to include the diffractive contributions from orbits which are nearly tangent to a concave part of the boundary. The leading correction for an unstable isolated orbit is of the…
We consider the one-person game of peg solitaire on a triangular board of arbitrary size. The basic game begins from a full board with one peg missing and finishes with one peg at a specified board location. We develop necessary and…
Using Marden's Theorem from geometric theory of polynomials, we show that for every triangle there is a unique ellipse such that the triangle is a billiard trajectory within that ellipse. Since $3$-periodic trajectories of billiards within…
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…
There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic,…
The Poincar\'e problem is a model of two-dimensional internal waves in stable-stratified fluid. The chess billiard flow, a variation of a typical billiard flow, drives the formation behind and describes the evolution of these internal…
Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a…
We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards…
We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these…
Lensed billiards are an extension of the notion of billiard dynamical systems obtained by adding a potential function of the form $C1_{\mathcal{A}}$, where $C$ is a real valued constant and $1_{\mathcal{A}}$ is the indicator function of an…
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…
By making use of the greatest common divisor's ($gcd$) properties we can highlight some connections between playing billiard inside a unit square and the Fibonacci sequence as well as the Euclidean algorithm. In particular by defining two…
The constrained Dirichlet boundary value problem $\ddot x=f(t,x)$, $x(0)=x(T)$, is studied in billiard spaces, where impacts occur in boundary points. Therefore we develop the research on impulsive Dirichlet problems with state-dependent…
We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional polygonal billiards. In polygons, diffraction typically occurs at the boundary of a family of…
The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank…
In a hyperbolic polygon any finite collection of closed billiard trajectories can be assigned an average length function. In this paper, we consider the average length of the collection of cyclically related closed billiard trajectories in…
We give the asymptotic growth of the number of primitive periodic trajectories of a two dimensional dispersive billiard, when we prescribe their number of bounces on one of the obstacles.
Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its…