The Complete Hyperbolicity of Cylindric Billiards
نویسنده
چکیده
The connected configuration space of a so called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with specular reflections at the boundaries of the removed cylinders. It is proven here that under a certain geometric condition — slightly stronger than the necessary condition presented in [S-Sz(1998)] — a cylindric billiard flow is completely hyperbolic. As a consequence, every hard ball system is completely hyperbolic — a result strengthening the theorem of [S-Sz(1999)].
منابع مشابه
Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers
In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called cylindric scatterers) have been removed. We prove that every such system is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for the ergodic...
متن کاملHyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards
When considering hyperbolicity in multi-dimensional Hamiltonian sytems, especially in higher dimensional billiards, the literature usually distinguishes between dispersing and defocusing mechanisms. In this paper we give a unified treatment of these two phenomena, which also covers the important case when the two mechanisms mix. Two theorems on the hyperbolicity (i.e. non-vanishing of the Lyapu...
متن کاملHyperbolic Billiards on Surfaces of Constant Curvature
We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with positive Lyapunov exponents on the sphere and on the hyperbolic plane.
متن کاملTrack Billiards
We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the seg...
متن کاملHyperbolic Magnetic Billiards on Surfaces of Constant Curvature
We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of the billiard dynamics, and give lower estimation for the Lyapunov exponent. This extends our recent results for non-magnetic billiards on surfaces of constan...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008