Billiards with positive topological entropy

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We prove that billiard flows on strictly convex tables with a sufficiently small circular scatterer generically admit positive topological entropy. In particular, we show that billiard systems in non-concentric circular annuli have the same property for sufficiently small inner radii in both Euclidean and hyperbolic spaces. Moreover the number of orbits which join two given outer configuration points in less than n iterations of the billiard map increases exponentially fast in n.