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 dissipative diffeomorphisms in two real dimensions. Using the notion of non-uniform partial hyperbolicity, we identify what it means for such maps to be “unimodal.” Then we discuss how techniques from one-dimensional dynamics (namely, Denjoy lemma and Koebe distortion theorem) can be adapted to prove a priori bounds for this class of maps.

This is based on a joint work with S. Crovisier, M . Lyubich and E. Pujals.