Title: Convergence of renormalization (and Sullivan's program)
Speaker: Professor Artur Avila
Speaker Info: CNRS, IMPA, Clay
Since the work of Feigenbaum and Coullet-Tresser on universality in the period doubling bifurcation, it is been understood that crucial features of unimodal (one-dimensional) dynamics depend on the behavior of a renormalization (and infinite dimensional) dynamical system. hile the initial analysis of renormalization was mostly focused on the proof of existence of hyperbolic fixed points, Sullivan was the first to address more global aspects, starting a program to prove that the renormalization operator has a uniformly hyperbolic (hence chaotic) attractor.Date: Tuesday, May 26, 2009
Key to this program is the proof of exponential convergence of renormalization along suitable ``deformation classes'' of the complexified dynamical system. Subsequent works of McMullen and Lyubich have addressed many important cases, mostly by showing that some fine geometric characteristics of the complex dynamics imply exponential convergence.
This contrasts with the original approach proposed by Sullivan, which centered around considerably rougher aspects of the complex dynamics (``precompactness''). We will describe recent work (joint with Lyubich) showing that exponential convergence does follow from precompactness features (by an abstract analysis of holomorphic iteration in deformation spaces), which enables us to conclude exponential convergence in all cases.