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For any simply connected polygon in the plane, the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections. This implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially. The main technical device is a calculation showing that the topological entropy of the Poincaré map for the billiard flow is equal to zero.

The growth rate for the number of singular and periodic orbits for a polygonal billiard