Lecture Abstracts

The Mandelbrot set M is a bifurcation diagram of the complex quadratic family. Its beautiful and rich structure continues to fascinate people.  In the beginning of the 80's, Douady and  Hubbard  undertook a deep  combinatorial  study of  this set which  explained many of its  features  observed on computer pictures. In particular, they developed a theory of little Mandelbrot copies inside M based on the theory of quadratic-like  maps. This theory laid down a foundation for the renormalization theory in the complex plane.

One of the central conjectures concerning the structure of M is that it is locally connected (MLC). On the one hand, this conjecture would complete a topological study of M including a proof of density of hyperbolic maps. On the other hand, it has deep connections with the Mostow-type rigidity phenomenon in hyperbolic geometry. Attempts to prove MLC led to many deep insights in holomorphic dynamics. The main analytical tools come from the theory of conformal invariants and quasi-conformal maps.

It is a beautiful combinatorial game discovered by Branner & Hubbard in the context of cubic maps and then transferred by Yoccoz to the quadratic set up. The idea is to cut the Julia set or the Mandelbrot set into pieces, describe their combinatorics in dynamical terms and then study their conformal geometry. What makes this theory special compared to the usual Markov coding is the presence of the critical point. What makes it tractable is conformal invariance or quasi-invariance of different geometric parameters. It is a key to many problems, including the rigidity and renormalization problems.
 

The "Feigenbaum Universality Law" discovered about 20 years ago and heuristically explained by the Renormalization Conjectures has fascinated both physicists and mathematicians. Physically, it gives a prediction of the transition parameter from "laminar" to "turbulent" regimes. Mathematically, it happened to be a deep problem on the borderline of dynamics, analysis and geometry. Methods of holomorphic dynamics turned out to be of crucial importance. The "dynamical part" of the problem was handled in 85 - 95 in the works of Sullivan and McMullen. The "parameter part" was recently handled by the author.
 

The main problem of dynamics is to describe a typical behaviour of orbits for a typical system. It turned out to be a very hard problem even for innocently looking families like the real quadratic family. However, the following recent result gives a solution to the problem: Almost any real quadratic map has either an attracting cycle, or an absolutely continuous invariant measure. In the former ("regular") case, almost all orbits converge to the cycle. In the latter ("stochastic") case, almost all orbits are asymptotically equidistributed with respect to the measure. The proof of this result incorporates all the theory mentioned above, and more.