On the Bernoulli Property of Planar Hyperbolic Billiards

We consider billiards in non-polygonal domains of the plane with boundaryconsisting of curves of three different types: straight segments, strictly convex inwardcurves and strictly convex outward curves of a special kind. The billiard map for thesedomains is known to have non-vanishing Lyapunov exponents a.e. provided that thedistance between the curved components of the boundary is sufficiently large, and the setof orbits having collisions only with the flat part of the boundary has zero measure. Undera few additional conditions, we prove that there exists a full measure set of the billiardphase space such that each of its points has a neighborhood contained up to a zero measureset in one Bernoulli component of the billiard map. Using this result, we show that thereexists a large class of planar hyperbolic billiards that have the Bernoulli property. This class includes the billiards in convex domains bounded by straight segments and strictlyconvex inward arcs constructed by Donnay.