Rounded triangular billiards

Fifty years ago, Leonid Bunimovich showed that convexbilliards could be hyperbolic, using billiards where the boundary consistsof straight segments and circular arcs such that the disks defined bythe arcs are contained in the billiard. Two well known examples withC1 boundaries, the straight and tilted stadia, have families of neutralperiodic orbits, and hence are not fully hyperbolic. We present theparameter space of the fully hyperbolic Bunimovich billiards formedby rounding the corners of the three integrable triangular billards. Byconstruction these have C1 boundaries and are convex.