On a fundamental theorem in the theory of dispersing billiards

Billiards are considered within domains in the plane or on the twodimensional torus with the euclidian metric, where the boundaries of these domains are everywhere convex inward. It is shown that the flow {S_t} generated by such a billiard is a K-system. A fundamental place is here assigned to the proof of the theorem showing that transversal fibers for the flow {S_t} consist “on the whole” of sufficiently long regular segments. From this theorem follow assertions on the absolute continuity of transversal fibers for the billiards in question.