DYNAMICS OF SOME PIECEWISE SMOOTH FERMI-ULAM MODELS

Abstract. We find a normal form which describes the high energy dynamics of a class of piecewise smooth Fermi-Ulam ping pong models; depending on the value of a single real parameter, the dynamics can be either hyperbolic or elliptic. In the first case we prove that the set of orbits undergoing Fermi acceleration has zero measure but full Hausdorff dimension. We also show that for almost every orbit the energy eventually falls below a fixed threshold. In the second case we prove that, generically, we have stable periodic orbits for arbitrarily high energies, and that the set of Fermi accelerating orbits may have infinite measure. 1. History and introduction In this paper we study the dynamics of piecewise smooth Fermi-Ulam ping pongs; Fermi and Ulam introduced such systems as a simple mechanical toy model to explain the occurrence of highly energetic particles coming from outer space and detected on Earth (the so-called cosmic rays, see [18, 19]). The model describes the motion of a ball bouncing elastically between a wall that oscillates periodically and