Tangent bundles dynamics and its consequences

We will consider here some dynamics of the tangent map, weaker than hyperbolicity, and we will discuss if these structures are rich enough to provide a good description of the dynamics from a topological and geometrical point of view. This results are useful in attempting to obtain global scenario in terms of generic phenomena relative both to the space of dynamics and to the space of trajectories. Moreover, we will relate these results with the study of systems that remain globally transitive under small perturbations.