A diffeomorphism of the disk is called mildly dissipative if, for every invariant measure, the stable manifold of almost every point disconnects the domain. Crovisier, Pujals and Tresser proved that every mildly dissipative diffeomorphism with zero topological entropy that is not generalized Morse–Smale is infinitely renormalizable.

In this talk, we survey a various regularity conditions under which such systems converge, under renormalization, to the universal renormalization attractor in the space of unimodal maps. We also discuss several ongoing projects and open directions related to this program.

This talk is based on joint work with Sylvain Crovisier, Mikhail Lyubich, and Enrique Pujals.