Positive Lyapunov Exponents versus Integrability in Random Conservative Dynamics

We study random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of systems, we establish an invariance principle stating that if all 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 consider random additive perturbations of the billiard map associated with a strictly convex planar table on a surface of constant curvature. In this setting, we show that the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. Second, we study random additive perturbations of a standard map and prove that the Lyapunov exponents vanish almost everywhere if and only if the map is integrable.