Partially hyperbolic diffeomorphisms: absolute continuity, rigidity, Lyapunov exponents

Abstract

We use new approaches and tools to better analyze the developments of some already obtained results related to rigidity of a partially hyperbolic diffeomorphism. These results, briefly, are examples of volume preserving Anosov on the three-dimensional torus such that the disintegration of volume on the center leaves is neither atomic nor Lebesgue; and examples of partially hyperbolic diffeomorphisms such that the center foliation is absolutely continuous but the diffeomorphism is not smoothly conjugate to its linearization. Such behavior does not coincide with the behavior observed on a recent result of Avila, Viana, Wilkinson [AVW] for perturbations of certain skew-products and time-one of an Anosov flow. The proposal of this new project, besides the understanding of the behavior of these conditional measures in the above examples, is to go further on the study of rigidity. On [AVW], absolute continuity of the center foliation, is equivalent to rigidity. Therefore, the example cited above having absolutely continuous center foliation and not smooth conjugation to its rigid model, reveals the need to impose stronger conditions on the center foliation to obtain rigidity. These are also related to group actions. (AU)