  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.
