Chaotic Properties of Billiards in Circular Polygons

We study billiards in domains enclosed by circular polygons. These are closed C 1 strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories close enough to the boundary of the domain, in which the return billiard dynamics is semiconjugate to a transitive subshift on infinitely many symbols that contains the full N -shift as a topological factor for any N ∈ N , so it has infinite topological entropy. We prove the existence of uncountably many asymptotic generic sliding trajectories approaching the boundary with optimal uniform linear speed, give an explicit exponentially big (in q ) lower bound on the number of q -periodic trajectories as q → ∞ , and present an unusual property of the length spectrum. Our proofs are entirely analytical.