Christian Bonatti (CNRS und Université de Bourgogne, Dijon, FRANKREICH)
Title :"Towards a global view of dynamical systems, for the C1-topology".

Abstract: In the beginning of the sixties, it appeared clearly that
chaotic dynamical systems could be important for many scientific
fields. At the same time, the hyperbolic theory of Anosov Smale
presents a large class of chaotic dynamics which could be completely
described and understood.
However, it is known from the last 60ties that there are open set of
non-hyperbolic dynamical systems : each of this system present fragil
behavior, so that there is no hope of a complete classification.
Results of many dynamicists in the last 10 years  suggest a program
for getting a global view of the dynamics of diffeomorphisms, from the
point of view of the C1-topology. More precisely, given any compact
manifold  M,  one splits the space Diff^1(M) in disjoint open regions
whose union is C1-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  C1-generic
dynamics.