相关论文: Birkhoff billiards are insecure
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…
A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics…
We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric…
In standard (mathematical) billiards a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite size hard…
Dynamical billiards, or the behavior of a particle traveling in a planar region $D$ undergoing elastic collisions with the boundary, has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of…
It is proven that, under some conditions on $f$, the non-compact flat billiard $\Omega = \{ (x,y) \in \R_0^{+} \times \R_0^{+};\ 0\le y \le f(x) \}$ has no orbits going {\em directly} to $+\infty$. The relevance of such sufficient…
Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then…
The billiard problem concerns a point particle moving freely in a region of the horizontal plane bounded by a closed curve $\Gamma$, and reflected at each impact with $\Gamma$. The region is called a `billiard', and the reflections are…
We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these…
In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist…
In the class of projective billiards, which contains the usual billiards, we exhibit counter-examples to Ivrii's conjecture, which states that in any planar billiard with smooth boundary the set of periodic orbits has zero measure. The…
We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential…
Given a strictly convex domain $\Omega$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we…
In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer…
A periodic trajectory on a polygonal billiard table is stable if it persists under any sufficiently small perturbation of the table. It is a standard result that a periodic trajectory on an $n$-gon gives rise in a natural way to a closed…
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…
The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.
We show a local rigidity result for the integrability of symplectic billiards. We prove that any domain which is close to an ellipse, and for which the symplectic billiard map is rationally integrable must be an ellipse as well. This is in…
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 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…