Related papers: Cosmological billiards and oxidation
We study the quantal energy spectrum of triangular billiards on a spherical surface. Group theory yields analytical results for tiling billiards while the generic case is treated numerically. We find that the statistical properties of the…
Cosmological billiards arise as a map of the solution to the Einstein equations, when the most general symmetry of the metric tensor is implemented, under the BKL (named after Belinskii, Khalatnikov and Lifshitz) paradigm, for which points…
We survey applications of the theory of hyperbolic (and to a lesser extent non hyperbolic) billiards to some fundamental problems of statistical physics and their mathematically rigorous derivations in the framework of classical Hamiltonian…
This article presents a new method to calculate eigenvalues of right triangle billiards. Its efficiency is comparable to the boundary integral method and more recently developed variants. Its simplicity and explicitness however allow new…
It is shown in detail that the dynamics of the Einstein-dilaton-p-form system in the vicinity of a spacelike singularity can be asymptotically described, at a generic spatial point, as a billiard motion in a region of Lobachevskii space…
We investigate the question whether there are cosmological models in 2+1 space-time dimensions which exhibit dynamics similar to BKL oscillations, as the cosmological singularity is approached. Based on intuition, we conceive a toy model…
This paper explores two instances where dissipation plays a crucial role in curbing the unbounded energy growth of particles in time-dependent billiards. The first example involves an elliptical-like billiard with inelastic collisions…
The classical inner and outer billiards can be formulated in variational terms, with length and area as the respective generating functions. The other two combinations, ``inner with area'' and ``outer with length,'' are more recently…
The properties of energy levels in a family of classically pseudointegrable systems, the barrier billiards, are investigated. An extensive numerical study of nearest-neighbor spacing distributions, next-to-nearest spacing distributions,…
We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and…
In this paper, we extend the scope of symbolic dynamics to encompass a specific class of ideal polyhedrons in the 3-dimensional hyperbolic space, marking an important step forward in the exploration of dynamical systems in non-Euclidean…
This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain $D\subset \mathcal R^3$. Two models will be analysed: in the first one, only an inner Keplerian…
We derive Cayley's type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We…
\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of…
The $p$-widths are a nonlinear analogue of the spectrum of the Laplacian. We prove that each $p$-width of a polygon in $\mathbb{R}^2$ is achieved by a union of billiard trajectories. We also compute the $p$-widths of the equilateral…
The classical dynamics of the isotropic two-dimensional harmonic oscillator confined by an elliptic hard wall is discussed. The interplay between the harmonic potential with circular symmetry and the boundary with elliptical symmetry does…
In this paper, we analyse the Einstein and Einstein-Maxwell billiards for all spatially homogeneous cosmological models corresponding to 3 and 4 dimensional real unimodular Lie algebras and provide the list of those models which are chaotic…
The purpose of this article is to highlight the fascinating, but only very incompletely understood relation between Einstein's theory and its generalizations on the one hand, and the theory of indefinite, and in particular hyperbolic, Kac…
From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve in the boundary of a smooth convex body can be approximated by a sequence…
In this paper we show that, under certain generic conditions, billiards on ovals have only a finite number of periodic orbits, for each period, all non-degenerate and at least one of them is hyperbolic. Moreover, the invariant curves of two…