Event № 71
Renormalization is a central idea of contemporary Dynamical Systems Theory, It allows one to control small scale structure of certain classes of systems, which leads to universal features of the phase and parameter spaces. We will review several occurrences of Renormalization in Holomorphic Dynamics: for quadratic-like, Siegel, and parabolic maps that enlighten the structure of many Julia sets and the Mandelbrot set. In particular, these ideas helped to construct examples of Julia sets of positive area (resolving a classical problem in this field). First examples were constructed by Buff and Cheritat about 10 years ago, and more recently a different class, with some interesting new features, was produced by Avila and the author. In the talk, we will describe these developments.