| Résume | I will focus on surface diffeomorphisms with zero entropy: Can the dynamics of these 'simple' systems be described? How does it bifurcate to positive entropy systems? These questions will be discussed for a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. It includes the dynamics of any Hénon diffeomorphism with Jacobian smaller than 1/4.
First, we will try to explain a joint work with Sylvain Crovisier and Charles Tresser that proved that any Hénon map with zero entropy and Jacobian smaller than 1/4 "is renormalizable" from a topological point of view.
Later I will discuss a work in progress with Sylvain Crovisier, Jonguk Yang and Misha Lyubich where we try to extend the topological renormalization to a differentiable one that could help to understand how the dynamics changes under small smooth perturbations. The proof is based on an "axiomatization of unimodal surface diffeomorphisms" that in particular, requires to develop the notion of "critical point" for surface diffeos. |
