Absolutely singular dynamical foliations

We show that for the C^1-open set of partially hyperbolic diffeomorphisms constructed in (M. Shub and A. Wilkinson, "Pathological foliations and removable zero exponents," Invent. math. 139 (2000) 3, 495-508), Lebesgue measure on the 3-torus decomposes as atomic measure along the leaves of the central foliation.Comment: 9 pages. See also http://www.math.nwu.edu/~wilkins

Centralizers of C^1-generic diffeomorphisms

On the one hand, we prove that the spaces of C^1 symplectomorphisms and of C^1 volume-preserving diffeomorphisms both contain residual subsets of diffeomorphisms whose centralizers are trivial. On the other hand, we show that the space of C^1 diffeomorphisms of the circle and a non-empty open set of C^1 diffeomorphisms of the two-sphere contain dense subsets of diffeomorphisms whose centralizer has a sub-group isomorphic to R

The centralizer of a C1 generic diffeomorphism is trivial

In this announcement, we describe the solution in the C1 topology to a question asked by S. Smale on the genericity of trivial centralizers: the set of diffeomorphisms of a compact connected manifold with trivial centralizer residual in Diff^1 but does not contain an open and dense subset

H\"older foliations, revisited

We investigate transverse H\"older regularity of some canonical leaf conjugacies in partially hyperbolic dynamical systems and transverse H\"older regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.Comment: 52 pages, to appear in Journal of Modern Dynamic