Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3D torus

The K-mixing property is proved for the simplest, non-trivial semi-dispersing billiard: that on the 3D torus with two cylindric scatterers (systems of elastic hard spheres can be represented as higher-dimensional toric billiards with cylindric scatterers). They also provide a method for a stronger, topological description of a constructively defined zero-measure set of points not necessarily belonging to open ergodic components because only this set could separate the ergodic components.