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

Source abstract

Let f f be a C 2 C^2 partially hyperbolic diffeomorphisms of T 3 \mathbb {T}^3 (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism A A with eigenvalues λ s > 1 > λ c > λ u . $$\begin{equation*} \lambda _{s}>1>\lambda _{c}>\lambda _{u}. \end{equation*}$$ Under the assumption that the set { x : ∣ log ⁡ det ( T f ∣ E c u ( x ) ) ∣ ≤ log ⁡ λ u } $$\begin{equation*} \{x: \,\mid \log \det (Tf\mid _{E^{cu}(x)})\mid \leq \log \lambda _{u} \} \end{equation*}$$ has zero volume inside any unstable leaf of f f where E c u = E c ⊕ E u E^{cu} = E^c\oplus E^u is the center unstable bundle, we prove that the stable foliation of f f is C 1 C^1 robustly minimal, i.e., the stable foliation of any diffeomorphism C 1 C^1 sufficiently close to f f is minimal. In particular, f f is robustly transitive. We build, with this criterion, a new example of a C 1 C^1 open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.