Positive Lyapunov Exponents versus Integrability in Random Conservative Dynamics

TL;DR

This paper investigates the relationship between positive Lyapunov exponents and integrability in random volume-preserving dynamical systems, establishing conditions under which exponents vanish and systems are integrable.

Contribution

It introduces an invariance principle linking zero Lyapunov exponents to invariant measures and applies it to billiard and standard maps to characterize integrability.

Findings

Lyapunov exponents vanish iff the billiard table is a geodesic disk.

Lyapunov exponents vanish iff the standard map is integrable.

Invariance principle connects zero exponents with invariant measures.

Abstract

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.