BILLIARDS IN THE L(P) UNIT BALLS OF THE PLANE

We study a one-parameter family of billiard maps T-p given by the conv ex tables \x\(p) + \y\(p) = 1 for p is an element of [1, infinity]. We note that the topological entropy is positive if p is not an element of {1, 2, infinity}. We study the linear stability of some periodic or bits and observe that the stability of these orbits changes for p = 2. For p is not an element of {1, 2, infinity}, there exist elliptic per iodic orbits suggesting that T-p is not ergodic for all p. We compute numerically the metric entropy of the maps T-p in the interval [1, 12] and the limiting behavior near p = 2.