Related papers: Outer Billiards, Digital Filters and Kicked Hamilt…
L. Boltzmann proposed a billiard model with a planar central force problem reflected against a line not passing through the center. He asserted that such a system is ergodic, which thus illustrates his ergodic hypothesis. However, it has…
Orbits in different dispersive billiard systems, e.g. the 3 disk system, are mapped into a topological well ordered symbol plane and it is showed that forbidden and allowed orbits are separated by a monotone pruning front. The pruning front…
An existence of an aperiodic point for outer billiard outside regular dodecagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic, and all possible periods are listed explicitly. The proof is…
For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for,…
A submanifold of the standard symplectic space determines a partially defined, multi-valued symplectic map, the outer symplectic billiard correspondence. Two points are in this correspondence if the midpoint of the segment connecting them…
We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an $O(\delta)$-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the…
A planar dual billiard is a planar curve $\gamma$ equipped with a family $(\sigma_P)|_{P\in\gamma}$ of projective involutions of the projective lines $L_P$ tangent to $\gamma$ at $P$ that fix $P$. A dual billiard is called rationally…
In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism,…
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto…
We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the…
This paper is the third in a series which explores a combinatorial method for generating lattice polygons in the plane. I call this method the plaid model. In this paper I prove the main result I had been aiming for since the beginning,…
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…
Transcendental H\'enon maps are the natural extensions of the well investigated complex polynomial H\'enon maps to the much larger class of holomorphic automorphisms. We prove here that transcendental H\'enon maps always have non-trivial…
We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special…
We present a computational scheme based on classical molecular dynamics to study chaotic billiards in static external magnetic fields. The method allows to treat arbitrary geometries and several interacting particles. We test the scheme for…
It is known that $C^1$-smooth strictly convex Radon norms in $\mathbb{R}^2$ can be characterized by the property that the outer billiard map, which corresponds to the unit ball of the norm, has an invariant curve consisting of 4-periodic…
We consider magnetic billiards under a strong constant magnetic field. The purpose of this paper is two-folded. We examine the question of existence of polynomial integral of billiard magnetic flow. We succeed to reduce this question to…
We develop a deformation-based framework for analyzing static billiard systems through the Jacobian determinant computed in noncanonical angular coordinates. Although these systems are conservative, the determinant is not identically equal…
The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each…
Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant…