Dynamics of surface diffeomorphisms with positive entropy

The dynamics of surface diffeomorphisms whose topological entropy vanishes is highly restricted: this is reflected for instance through works by Franks-Handel, Le Calvez-Tal, Crovisier-Pujals-Tresser,… For surface diffeomorphisms whose entropy is positive, the dynamics seems much more flexible, but satisfies also some constrains: for instance it always contains a horseshoe (Katok’s theorem).

In this talk I will discuss an attempt we develop with E. Pujals for describing the dynamics of general surface diffeomorphisms which are dissipative, focusing on their decomposition.