Generalized relativistic billiards

We study generalized relativistic billiards, which is the following dynamical system. A particle moves in the interior of a domain under the influence of some force fields. As the particle hits the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall. Both the motion in the domain and the reflection are considered in the framework of the theory of relativity.
We study the periodic and "monotone" action of the boundary for the particle moving in a parallelepiped and in an arbitrary compact domain respectively, and we also consider an "accelerating" model in an unbounded domain. We prove that under some general conditions an invariant manifold in the velocity phase space of the generalized billiard, where the particle velocity equals the velocity of light, either is an exponential attractor or contains one. Thus for an open set of initial conditions the particle energy tends to infinity.