Lyapunov exponents for small random perturbations of predominantly hyperbolic two dimensional volume-preserving diffeomorphisms, including the Standard Map

An outstanding problem in smooth ergodic theory is the estimation from below of Lyapunov exponents for maps which exhibit hyperbolicity on a large but non- invariant subset of phase space. It is notoriously difficult to show that Lypaunov exponents actually reflect the predominant hyperbolicity in the system, due to cancellations caused by the“switching” of stable and unstable directions in those parts of phase space where hyperbolicity is violated. In this talk I will discuss the inherent difficulties of the above problem, and will discuss recent results when small IID random perturbations are introduced at every time-step. In this case, we are able to show with relative ease that for a large class of volume-preserving predominantly hyperbolic systems in two dimensions, the top Lypaunov exponent actually reflects the predominant hyperbolicity in the system. Our results extend to the well-studied Chirikov Standard Map with large coupling coefficient. This work is joint with Lai-Sang Young and Jinxin Xue.