SURVEY Towards a global view of dynamical systems, for the C1-topology

Abstract This paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C 1 -topology. More precisely, given any compact manifold M , one splits Diff 1 ( M ) into disjoint C 1 -open regions whose union is C 1 -dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: • either a robust local phenomenon; • or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C 1 -generic dynamics.