These diffeomorphisms exhibit weaker forms of hyperbolicity and are extremely common. Such a diffeomorphism f has stable, unstable and center bundles invariant under df. This is an very intense area of research currently. We review basic examples, conjectures.

We also talk about dynamical coherence - this has to do with the integrability of the center stable, center unstable bundles. Unlike the strictly hyperbolic case, there are non integrable examples. This leads to some recent counterexamples of a main conjecture.