Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems

We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C ∞ Riemannian metric whose geodesic flow is Bernoulli.