Daniel Smania: Renormalization Theory (Mini Course)

We will give an introduction to renormalization theory in one-dimensional dynamics, most of the time keeping ourselves in the study of the Feigenbaum case and using simplifications introduced by the author in McMullen and Lyubich work in this case. We intend to explain the hyperbolicity of the Feigenbaum fixed point and its consequences.

Feigenbaum and Collet-Tresser, in the 70's, noted that certain geometrical and topological aspects of the bifurcation diagram of families of unimodal maps (maps of the interval with an unique critical point) were universal: it do not depend on the particular family studied. They conjectured that such universality could be explained by the existence of a hyperbolic fixed point (the so-called Feigenbaum fixed point) for a operator called renormalization operator, which is defined on the space of unimodal maps: so certain aspects of one-dimensional dynamics could be explained studying the dynamics of a operator on an infinite-dimensional space!

Lanford gave a proof of the main part of Feigenbaum conjecture on 80's. Such proof was one of the first computer-assisted proofs in analysis. However, this impressive result was unable to dissipate the insterest of mathematicians for a more conceptual proof, once the Lanford proof itself did not bring much light on the universality fenomena. The following major step for a conceptual proof of Feigenbaum conjecture was given by D. Sullivan on the middle 80's: he indeed came with a totally new setting of tools to deal with the problem: after Sullivan and contributions of Douady \& Hubbard, most of renormalization theory became a solid subfield of complex dynamics, where methods using quasiconformal maps, introduced by Sullivan himself in the study of complex dynamics, are one of the main tools. Sullivan gave a conceptual proof of the existence and uniqueness of the Feigenbaum fixed point and a characterization of its "stable manifold".

C. McMullen, in the 90's, obtained a much simpler proof of the existence and uniqueness of the Feigenbaum fixed point, and proved that the renormalizations of the maps in the "stable manifold" indeed converges exponentially fast to the fixed point (in Sullivan result, there was not a estimative of how fast the renormalizations converges to the Feigembaum fixed point).

Finally, M. Lyubich, in a major break throught, gave a conceptual proof of the hyperbolicity of the Feigenbaum fixed point. The Lyubich argument reduces the hyperbolicity of the Feigenbaum fixed point to a rigidity result: rigidity problems are a central issue in complex dynamics, since many problems in it can be translated to rigidity problems.

In the mini-course, we are going to give a short introduction (so complete as the time allows us) to the methods used on the field, keeping ourselves in the simplest situation possible: the Feigenbaum renormalization. In the first exposition we intend to begin with an introduction suitable to everyone familiar with basic notions on dynamics. Then we will go deeper in complex dynamics methods to prove the hyperbolicity of the Feigenbaum fixed point. Finally, in the last exposition(s) we will give an introduction to generalized renormalization, in particular to the Fibonacci renormalization, and state new results proved by us in this setting.