Some hyperbolicity revisited and robust transitivity

In this article we revisit the notion of $\mathit{\text{Some}}\mathit{\text{Hyperbolicity}}$ introduced by Pujals and Sambarino in \cite{PuSa}. We present a more general definition, that in particular can be applied to the symplectic context (something that the previous couldn’t). As an application we construct $C^1$ robustly transitive derived from Anosov diffeomorphisms with mixed behaviour on center leaves.