Convergence of renormalization

14 May 2010
Professor Artur Avila
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. While 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. 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. We will describe recent work (joint with Lyubich) which moves the focus to the abstract analysis of holomorphic iteration in deformation spaces. It shows that exponential convergence does follow from rougher aspects of the complex dynamics (corresponding to precompactness features of the renormalization dynamics), which enables us to conclude exponential convergence in all cases.