I will motivate the study of conservative diffeomorphisms isotopic to Anosov on T^3 with a weak (two-bundle) partially hyperbolic splitting. Despite its simple setting, such diffeomorphisms are likely to exhibit many interesting and little understood phenomena. Perhaps the most fundamental question is whether all or most such diffeomorphisms are ergodic. Indeed very little is known about ergodicity outside the traditional setting of three-bundle partial hyperbolicity. In this talk I will present a result on the much easier problem of transitivity.
This is a joint work with Shaobo Gan.