Related papers: Diffractive orbits in isospectral billiards

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space.…

This paper is a follow-up on the \emph{noncommutative differential geometry on infinitesimal spaces} [15]. In the present work, we extend the algebraic convergence from [15] to the geometric setting. On the one hand, we reformulate the…

We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $\mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each…

We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we…

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary…

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…

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads…

We present a functional analytic framework based on the spectrum of the transfer operator to study billiard maps associated with perturbations of the periodic Lorentz gas. We show that recently constructed Banach spaces for the billiard map…

A real Liouville domain is a Liouville domain together with an exact anti-symplectic involution. We call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically an invariant…

We compute the spectrum of the Feigenbaum period-doubling operator in the space of bounded analytical functions in an ellipse. The spectral properties of the period-doubling operator in this space are not the same as in the space of even…

We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…

We show that wave functions in planar rational polygonal billiards (all angles rationally related to Pi) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly…

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…

The paper establishes the property of splittability of billiard boundary sequences in n dimensional cube into subsequences of fractional parts. This reveals a new property of integrable and weak perturbated Hamilton systems: under a simple…

We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they…

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 a multi-dimensional billiard system in an (n+1)-dimensional Euclidean space, the direct product of the "horizontal" hyperplane and the "vertical" line. The hypersurface that determines the system is assumed to be smooth and…

A statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards, one with mushroom-like shape and the other from the familiy of the Limacon billiards, is presented. These billiards have mixed…

Euclidean outer billiard on a regular polygon (that is not a triangle, square or a hexagon) has aperiodic points, i.e., points where all iterates of the outer billiard map are defined and yield pairwise distinct images. This result answers…

Motivated by the high-energy limit of the $N$-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a…