ABSTRACT. Given a surface M and a Borel probability measure ν on the group of C 2 -diffeomorphisms of M , we study ν-stationary probability measures on M . Assuming the positivity of a certain entropy, the following dichotomy is proved: either the stable distributions for the random dynamics is non-random, or the measure is SRB. In the case that ν-a.e. diffeomorphism preserves a common smooth measure m, we show that for any positiveentropy stationary measure µ either there exists a ν-almost-surely invariant µ-measurable line field (corresponding do the stable distributions for a.e. random composition) or the measure µ is ν-almost-surely invariant and coincides with an ergodic component of m.To prove the above result, we introduce a skew product with surface fibers over a measure preserving transformation equipped with an increasing sub-σ-algebraF . Given an invariant measure µ for the skew product, and assuming theF -measurability of the 'past dynamics' and the fiber-wise conditional measures, we prove a dichotomy: either the fiberwise stable distributions are measurable with respect to a related increasing sub-σ-algebra, or the measure µ is fiber-wise SRB.

We prove that stable ergodicity is C r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r ∈ [2, ∞].The proof follows Pugh-Shub program [21]: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C r open and dense, and essential accessibility implies ergodicity.

Abstract. Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7) Transitivity and (8) Classification. Here we will survey the state of the art on these subjects, and propose related problems.

We find a class of ergodic linear automorphisms of T N that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of T N are stably ergodic when N ≤ 5.

Abstract. We consider an ergodic invariant measure µ for a smooth action α of Z k , k ≥ 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R k , k ≥ 2 on a (2k + 1)-dimensional manifold. We prove that if µ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z k has positive entropy, then µ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.