A lower bound for chaos on the elliptical stadium.

Abstract

The elliptical stadium is a plane region bounded by a curve constructed by joining two half-ellipses, with half axes a > 1 and b = 1, by two parallel segments of equal length 2h. Donnay [Comm. Math. Phys. 141 (1991) 225–257] proved that if 1 < a < 2 and if h is large enough then the corresponding billiard map has non-vanishing Lyapunov exponents almost everywhere; moreover h → ∞ as a → 2. In a previous paper [Markarian et al. Comm. Math. Phys. 174 (1996) 661–679] we found a bound for h assuring the K-property for these billiards, for values of a very close to 1. In this work we study the stability of a particular family of periodic orbits obtaining a new bound for the chaotic zone for any value of a <2.}