Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

Abstract

We consider a large class of 2D area-preserving diffeomorphisms that arenot uniformly hyperbolic but have strong hyperbolicity properties on large regions oftheir phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potentialswitching of ``stable" and ``unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains withrelative ease Lyapunov exponents reflecting the geometry of the deterministic maps.