| 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. In particular, we will prove that any Hénon map with zero entropy and Jacobian smaller than 1/4 "can be renormalized". As a consequence, we obtained a two-dimensional version of Sharkovsky's theorem about the set of periods of interval maps. This is part of a joint work with S. Crovisier and C. Tresser If the time permits we will discuss a work in progress with Sylvain Crovisier, Misha Lyubich and Jonguk Jang about the bounded geometry of those renormalizable systems. |
