On a fundamental theorem in the theory of dispersing billiards

Billiards are considered within domains in the plane or on the two-dimensional 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.
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Bibliography: 6 titles.