Bernoulli Diffeomorphisms on Surfaces

The smooth ergodic theory deals with the metric (ergodic) properties of classical dynamical systems (i.e., diffeomorphisms and smooth one-parameter flows on smooth manifolds) with respect to a positive smooth invariant measure. By a positive smooth measure we mean a measure on a smooth (usually Co) manifold (possibly with boundary) which is given by a positive smooth density in every local coordinate system. We shall say that a diffeomorphism f: M -> M is a Bernoulli diffeomorphism if (1) f preserves some smooth positive probability measure p on M and (2) considered as an automorphism of the Lebesgue space (M, p), the mapping f is metrically isomorphic to a Bernoulli shift. Recently Ja. B. Pesin (1, 21, [3) established remarkable connections between the Lyapunov characteristic exponents and the ergodic properties of classical dynamical systems on compact manifolds. One of the main results of Pesin can be described as follows. Almost every ergodic component of a diffeomorphism f: M-> M with respect to a smooth invariant measure, belonging to the set