Robust minimality of strong foliations for DA diffeomorphisms: cu-volume expansion and new examples

Let f be a C2 partially hyperbolic diffeomorphisms of T3 (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism A with eigenvaluesλs<1<λc<λu.Under the assumption that the set{x:∣log⁡det⁡(T⁢f∣Ec⁢u⁡(x))∣≤log⁡λu}has zero volume inside any unstable leaf of f where Ec⁢u=Ec⊕Eu is the center unstable bundle, we prove that the stable foliation of f is C1 robustly minimal, i.e., the stable foliation of any diffeomorphism C1 sufficiently close to f is minimal. In particular, f is robustly transitive.

We build, with this criterion, a new example of a C1 open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.