  We are interested in finding a dense part of the space of
$C^1$-diffeomorphisms which decomposes into open subsets corresponding to
different dynamical behaviors: we discuss results and questions in this
direction. In particular we present recent results towards a conjecture by J.
Palis: any system can be approximated either by one which is hyperbolic (and
whose dynamics is well understood) or by one which exhibits a homoclinic
bifurcation (a simple local configuration involving one or two periodic
orbits).
