Related papers: A Random Billiard Map in the Circle
The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensional barrier billiards. These random matrices are extracted…
We study a class of elliptic billiards with a Keplerian potential inside, considering two cases: a reflective one, where the particle reflects elastically on the boundary, and a refractive one, where the particle can cross the billiard's…
The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and…
We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument,…
In this paper we prove that in any analytic one-parameter family of twist maps of the annulus, homotopically invariant curves filled with periodic points corresponding to a given rotation number, either exist for all values of the…
We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not…
The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires…
A circular Andreev billiard in a uniform magnetic field is studied. It is demonstrated that the classical dynamics is pseudointegrable in the same sense as for rational polygonal billiards. The relation to a specific polygon, the asymmetric…
Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In…
We consider a random walk in a stationary ergodic environment in $\mathbb Z$, with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right…
We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated…
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 stability of periodic trajectories of planar inverse magnetic billiards, a dynamical system whose trajectories are straight lines inside a connected planar domain $\Omega$ and circular arcs outside $\Omega$. Explicit examples…
Finite horizon Sinai billiard maps are examples of uniformly hyperbolic systems with singularities. These discontinuities make it more difficult to develop the classical theory of thermodynamic formalism. Nevertheless, Baladi and Demers…
In order to verify Percival's conjecture [J. Phys. B 6,L229 (1973)] we study a planar billiard in its classical and quantum versions. We provide an evaluation of the nearest-neighbor level-spacing distribution for the Cassini oval billiard,…
The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. Gravitational billiards provide an experimentally accessible arena for their investigation. We present a mathematical model that captures the…
We review some properties of periodic orbit families in polygonal billiards and discuss in particular a sum rule that they obey. In addition, we provide algorithms to determine periodic orbit families and present numerical results that shed…
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.
We prove Knudsen's law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles. Dynamics are randomly perturbed and the corresponding random map studied under a skew-type…
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove…