相关论文: Escape Orbits for Non-Compact Flat Billiards

We prove that if $\Gamma$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing…

Let x and y be points in a billiard table M that is bounded by a curve sigma. We assume that sigma is a simple closed C^r curve with positive curvature, where r is at least 2. A subset B of M\{x,y} is called a blocking set for the pair…

We consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a…

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 correspondence between the orbits of a system of 2 complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general, in the sense that…

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to…

Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper function bounded from below. Suppose the sublevel set {H<d} contains the zero section and \alpha is a non-trivial homotopy class of free loops in M. Then for…

If D is a rational polygon, then the associated rational billiard table is given by \Omega(D). Such a billiard table is well understood. If F is a closed fractal curve approximated by a sequence of rational polygons, then the corresponding…

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected…

We introduce a class of convex, higher-dimensional billiard models which generalise stadium billiards. These models correspond to the free motion of a point-particle in a region bounded by cylinders cut by planes. They are motivated by…

We investigate the existence of elliptic islands for a special family of periodic orbits of a two-parameter family of maps corresponding to the billiard problem on the elliptical stadium. The hyperbolic or elliptical character of these…

Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has…

We consider open circular billiards with one and with two holes. The exact formulas for escape are obtained which involve the Riemann zeta function and Dirichlet L-functions. It is shown that the problem of finding the exact asymptotics in…

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the…

We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. We give new results on periodicity and boundedness of orbits which suggest…

We investigate the escape dynamics in an open circular billiard under the influence of a uniform gravitational field. The system properties are investigated as a function of the particle total energy and the size of two symmetrically placed…

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…

In this paper we show that in some cases the E.Hopf rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are sharp, i.e.…

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 construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability…