Related papers: Unbounded orbits for the Plaid Model
The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological well-ordered symbol plane. In the symbol plane the pruning front is…
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with…
The famous conjecture of V.Ya.Ivrii says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study its complex analytic version for…
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…
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…
Optical mushroom shaped billiards offer a unique opportunity to isolate and study non-dispersive, marginally unstable periodic orbits. Here we show that the openness of the cavity to external fields presents unanticipated consequences for…
We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table. Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in a self-similar Sierpinski…
In this work, we construct linearly stable periodic orbits in $3$-dimensional domains with boundaries containing focusing components (small pieces of a sphere) where we place these components arbitrarily far apart. It demonstrates that we…
Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years polygonal billiards have attracted great attention due to their application in the understanding of anomalous transport, but also at the fundamental level,…
We give a simple proof of our previous result with V. Zharnitsky that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a…
We prove that for a dynamical system on an algebraic variety over $\overline{\mathbb{Q}}$ generated by finitely many unramified endomorphisms, it is decidable whether a given point has a finite orbit. This is achieved by establishing an…
We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those…
A planar polygonal billiard $\P$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $\P$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to…
We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth…
In this text we study billiards on ovals and investigate some consequences of a rotational symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits…
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…
An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary, and also a circular scatterer in the interior of the disk. We investigate stability properties of some…
We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra $\mathfrak {gl}(n,m)$, depending on a parameter. We show that for generic values of the parameter all the orbits are…
We use Ratner's theorem to compute the asymptotics of the number of (cylinders of) periodic trajectories in a rectangle with a barrier, assuming that the location p/q of the barrier is rational. We also show that as q tends to infinity, the…
We propose geometric tools that are suitable for studying the behavior of a billiard trajectory in a homogeneous force field. Two examples are considered: a vertical plane with an open top and with a parabolic or right angle boundary at the…