Using integrability to produce chaos: Billiards with positive entropy (Q1181842)

In this paper the author studies the ergodic properties of billiards inside a planar domain \(Q\) whose boundary \(\partial Q\) consists of piecewise smooth arcs that are convex or flat. A billiard in \(Q\) is the dynamical system determined by uniform motion of a point mass in \(Q\) with elastic reflections at the boundary. The author introduces a very general class of convex arcs ( focusing arcs) for which the resulting billiard has positive Lyapunov exponents almost everywhere. Then, by Pesin's theorem, these dynamical systems have positive measure-theoretic entropy, and are chaotic. The class of convex arcs which the author constructs is open in the \(C^ 6\) topology on curves. Using his general results, the author then proves that small \(C^ 6\) perturbations of the Bunimovich stadium billiard have positive Lyapunov exponents.