BRIDGE TO HYPERBOLIC POLYGONAL BILLIARDS

The paper proves two main claims: (i) for any non-convex simply connected rational polygon P, there is r_P>0 such that the physical billiard is hyperbolic for all 0<r<r_P, by showing almost all orbits hit dispersing arcs infinitely often and invoking Sinai’s continued-fraction criterion for hyperbolicity, including a detailed statement of the positive-term continued fraction and the Seidel–Stern convergence condition (eq. (3.2)); and (ii) there is an open dense subset of all polygons such that, for sufficiently small r, the physical billiard is hyperbolic on a subset of positive measure of phase space, using a Poincaré-recurrence argument on a positive-measure set of states based on dispersing arcs . The candidate solution reaches the same type of generic positive-measure hyperbolicity via a different route: an induced first-return map to trimmed dispersing arcs inside a concave “bay,” a uniform cone-field expansion per return, and transfer to the full map. While the model imposes a more structured geometric configuration (two visible dispersing arcs) than the paper’s minimal “at least one reflex angle” hypothesis, this stricter condition is dense and ensures the claimed expansion mechanism. Minor technical gaps (e.g., explicit use of Kac’s lemma to pass from induced-map expansion to positive full-map Lyapunov exponent, and the uniform bound on the number of legs between returns) are readily addressed and do not contradict the paper. Hence, both are correct, by different proofs.