Sylvain Crovisier: Mildly dissipative maps of the disk with zero entropy III

Abstract:

These lectures 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?

One will discuss 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 (real) Hénon diffeomorphism with Jacobian smaller than 1/4.

One will then explain a joint work between Sylvain Crovisier, Enrique Pujals and Charles Tresser which asserts that any Hénon map with zero entropy and Jacobian smaller than 1/4 is “renormalizable” from a topological point of view.

In a work in progress with Jonguk Yang and Misha Lyubich 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 diffeomorphisms.