Inevitable escape
by
Maciej P. Wojtkowski
University of Arizona

We consider the problem of bounded orbits in hyperbolic systems which are open, i.e., in which escape is possible. It is natural to expect that in such systems escape is inevitable, i.e., almost all orbits escape. We will discuss a model problem of a discontinuous piecewise linear hyperbolic map in the plane. These ideas can be then applied to the system of perfectly elastic particles in a box with one of the walls moving periodically. We assume that the wall is infinitely heavy, so that it is not affected when colliding with the particles. The particles though may gain or loose energy as a result of the elastic collision with the moving wall. Thus the total energy of the system may increase unboundedly. Is it really the case? Hyperbolicity leads to the following answer: the set of all orbits on which the energy is bounded has zero Lebesgue measure. We will discuss conditions which give us such a theorem.

Date received: February 1, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaa-55.