Dynamics of $C^1$-diffeomorphisms: global description and prospects for classification

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).