Lyapunov Exponents versus Integrability in Random Conservative Dynamics

Abstract: We consider random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of random systems, we establish an invariance principle asserting that if the Lyapunov exponents vanish, then there exists a measurable family of probability measures on the projective bundle that is invariant under the projective cocycle induced by the derivative. We apply this principle to two classes of random systems. First, we study random additive perturbations of a single billiard map associated with a strictly convex planar table on a surface of constant curvature. In this setting, we prove that the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. Second, we consider random additive perturbations of a single standard map and show that the Lyapunov exponents vanish almost everywhere if and only if the standard map is integrable.