Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards

In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. We prove that such systems are ergodic and are $K$-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.