Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards

When considering hyperbolicity in multi-dimensional Hamiltonian sytems, especially in higher dimensional billiards, the literature usually distinguishes between dispersing and defocusing mechanisms. In this paper we give a unified treatment of these two phenomena, which also covers the important case when the two mechanisms mix. Two theorems on the hyperbolicity (i.e. non-vanishing of the Lyapunov exponents) are proven that are hoped to be applicable to a variety of situations. As an application we investigate soft billiards, that is, replace the hard core collision in dispersing billiards with disjoint spherical scatterers by motion in some spherically symmetric potential. Analogous systems in two dimensions have been widely investigated in the literature, however, we are not aware of any mathematical result in this multi-dimensional case. Hyperbolicity is proven under suitable conditions on the potential. This way we give a natural generalization of the hyperbolicity results obtained before in two dimensions for a large class of potentials.