An Infinite Step Billiard

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 $\Omega = \bigcup_{n\in\N} [n,n+1] \times [0,p_n]$, 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.