Wolff Lecture

Part 1:  Renormalization is an opearator that relates the dynamics in various scales. Good control of this operator usually implies remarkable dynamical consequences, e.g., the universal shape of little M-copies. However, it is a difficult problem to gain such a control. We will outline a proof of the Feigenabaum-Coullet-Tresser Renormalization Conjecture, based upon Sullivan's, McMullen's and author's work from the 1990s. 

Part 2:  There are two types of the dynamics observed in the real quadratic family: regular dynamics governed by a unique attracting cycle, and stochastic dynamics governed by a unique absolutely continuous invariant mesaure. We will outline a proof of the Regular or Stochastic Dichotomy that asserts that almost any real quadratic map z^2+c, c\in [-2, 1/4], is either regular or stochastic, which gives a complete probabilistic picture of the dynamics in this family. The proof exploits the Yoccoz Puzzle and the Renormalization machineries. Based upon author's work from the 1990s.