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 settingâ€”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.
Meeting ID: 852 4277 3200
Passcode: 103121