Simons Semester in Banach Center: "Dynamical Systems"

Renormalization is a strong tool to study the fine-scale structures in low-dimensional dynamics. Starting with a class of maps, to each f in the class, one often identifies an appropriate iterate of f on a region in its domain of definition, which, once viewed in a suitable coordinate on the region, belongs to the same class of maps. Remarkably, iterating a renormalization operator on a class of maps provides significant information about the behavior of individual maps in the class. Inou and Shishikura in 2006 introduced a renormalization scheme to study the local dynamics of holomorphic germs asymptotic to irrational rotations, and their perturbations. Although an strong tool to prove non-trivial results on the dynamics of such maps, the renormalization itself is based on a sophistication construction, and requires geometric control on some (canonically defined) transcendental changes of coordinates that have highly discoing nature. In a series of lectures, we plan to present a constructive approach to this renormalization scheme, and introduce a new analytic technique to control the geometric quantities that appear in the renormalization. As an application, we shall sketch the arguments behind a number of recent results obtained using this renormalization scheme.