An Infinite Step Billiard

Abstract

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e., systems of a point particle moving freely in the domain $\Omega ={\bigcup }_{n\in \mathbb{N}}[n,n+1]\times [0,p_n]$, with elastic reflections on the boundary; here $p_0=1,p_n>0$ and $p_n$ vanishes monotonically. After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example $p_n=2^{-n}$. What plays an important role in this case are the so called escape orbits, that is, orbits going to $+\infty$ monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard.

34 pages in LaTeX 2.09 including 8 ps figures (with psfig.tex)

References in corpus (1)

• Escape Orbits for Non-Compact Flat Billiards

Cited by in corpus (4)

• Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains
• Escape orbits and Ergodicity in Infinite Step Billiards
• Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces
• Indiscriminate covers of infinite translation surfaces are innocent, not devious