Dynamics of Renormalization Operators

Abstract

It is a remarkable characteristic of some classes of low-dimensional dynamical systems that their long time behavior at a short spatial scale is described by an induced dynamical system in the same class. The renormalization operator that relates the original and the induced transformations can then be iterated, and a basic theme is that certain features (such as hyperbolicity, or the existence of an attractor) of the resulting “dynamics in parameter space” impact the behavior of the underlying systems. Classical illustrations of this mechanism include the Feigenbaum-Coullet-Tresser universality in the cascade of period doubling bifurcations for unimodal maps and Herman’s Theorem on linearizability of circle diffeomorphisms. We will discuss some recent applications of the renormalization approach, focusing on what it reveals about the dynamics at typical parameter values.