Some hyperbolicity revisited and robust transitivity.

The author studies partially hyperbolic diffeomorphisms and introduces the SH-saddle property (for unstable and stable foliations, respectively), extending the SH property introduced in [E. R. Pujals and M. Sambarino, Ergodic Theory Dyn. Syst. 26, No. 1, 281–289 (2006; Zbl 1085.37022)]. Here, SH stands for some hyperbolicity and requires that every disk of radius 1 contained in a strong unstable leaf contains a point at which the central bundle exhibits uniformly expanding behavior. The weaker SH-saddle property instead characterizes this uniform expansion in the central bundle through a cone criterion. The author proves that the SH-saddle property is a C1-open condition within the class of partially hyperbolic diffeomorphisms. The paper also presents a criterion for DA diffeomorphisms that guarantees C1-robust transitivity and provides new DA examples.