  We consider a large class of 2D area-preserving diffeomorphisms that are not
uniformly hyperbolic but have strong hyperbolicity properties on large regions
of their 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
potential switching of "stable" and "unstable" directions. This paper shows
that with the addition of (very) small random perturbations, one obtains with
relative ease Lyapunov exponents reflecting the geometry of the deterministic
maps.
