Billiards in the lp unit balls of the plane

We study a one-parameter family of billiard maps Tp given by the convex tables ¦x¦p + ¦y¦p = 1 for p [1, ∞]. We note that the topological entropy is positive if p {1, 2, ∞}. We study the linear stability of some periodic orbits and observe that the stability of these orbits changes for p = 2. For p {1, 2, ∞}, there exist elliptic periodic orbits suggesting that Tp is not ergodic for all p. We compute numerically the metric entropy of the maps Tp in the interval [1, 12] and the limiting behavior near p = 2.