Absolute continuity of the central foliation for partially hyperbolic systems

Abstract

We shall discuss on this project the absolute continuity of the central foliation for partially hyperbolic diffeomorphisms. The starting point is the PhD thesis of the candidate which among other results provided the first example on $ \mathbb T^3 $ of a conservative "Derived from Anosov" with disintegration of the volume in the central leaves singular with respect to Lebesgue and non-atomic. We conjecture that all conservative Anosov on $\mathbb T ^ 3$ seen as partially hyperbolic ($ TM = E^s \oplus E^c \oplus E^u $) has non-atomic disintegration, the verification of this conjecture is one of the goals to this post-doctoral project. One result from the thesis of the candidate, showed the relationship between mensurability (in the sense of Rokhlin) and it's disintegration. But some more powerful tools are necessary for better understanding the non-mensurability of the foliation. The articles of Ledrappier, Young are the candidates for the first approach, by treating the case of non-compact foliation (which is not always mensurable) through what they called "subordinate partition". (AU)