On the one hand, the spectral theorem claims that the dynamics of hyperbolics systems can be decomposed into finitely many independent and elementary pieces(basic sets). On the other hand, there are systems exhibiting in a "persistent" way infinitely many pieces of dynamics (for instance, sinks); this is the so-called Newhouse phenomenon.

In the context of $C1$-generic dynamics, we discuss some results stating the dichotomy tame vs wild dynamics. Tame systems are those having finitely many elementary pieces of dynamics. Moreover, these systems satisfy some weak form of hyperbolicity and some of the properties of the hyperbolic systems. We also explain how that wild dynamics arises.