Entropy of billiard maps and a dynamical version of the Birkhoff conjecture

We first prove that the simplest (smooth convex) billiard table is the circular one, in the sense that the associated billiard map has polynomial entropy equal to 1, while for all other tables the billiard maps have polynomial entropy ≥2. We then prove that the billiard maps of noncircular elliptic tables have polynomial entropy equal to 2. This yields a natural entropic version of the classical Birkhoff conjecture: the elliptic tables are the only ones whose associated maps have polynomial entropy equal to 2.