One of the most fundamental examples of non-linear dynamics is given by the class of unimodal interval maps. It is the simplest setting in which one can study the behavior of a critical orbit and the profound impact it has on the geometry of the system. By the works of Sullivan, McMullen and Lyubich, we have a complete renormalization theory for these maps, and as a result, their dynamics is now very well understood.
In this talk, we discuss the extension of this theory to a higher dimensional setting -- namely, to properly dissipative diffeomorphisms in dimension two. Using the notion of non-uniform partial hyperbolicity, we identify what it means for such maps to be "unimodal". Then we show that properly dissipative infinitely renormalizable unimodal diffeomorphisms have a priori bounds (a certain uniform control on their geometry that holds at arbitrarily small scales).
This is based on a joint work with S.Crovisier, M.Lyubich and E.Pujals.