EMS LECTURES, 1999: Real and Complex Dynamics [ Back ]

Dates:
31.05.99 - 04.06.99
Place:
Barcelona, Spain
Organizers:
Núria Fagella
Web Site:
http://www.maia.ub.es/~fagella/L...

Speaker

Mikhail Lyubich (`European Mathematical Society Lecturer'' of 1999), State University of New York at Stony Brook, USA.

Abstract

Real and complex one-dimensional dynamics have been in the focus of research for the past 20 years. The ideas of dynamics, statistical physics, geometric function theory and hyperbolic geometry (supported by computer experiments) came together to reveal many rich and fascinating structures. As a result, several central problems have been recently resolved. Among them are the proofs of the real Rigidity Conjecture, the Feigenbaum Universality Conjecture and the Regular or Stochastic Conjecture. Altogether it gives a complete picture of dynamics in the real quadratic family.

Contents

  1. THE MANDELBROT SET AND ITS LITTLE COPIES
    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.
  2. RIGIDITY CONJECTURES AND THEOREMS
    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.
  3. PUZZLES AND PARAPUZZLES
    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.
  4. RENORMALIZATION CONJECTURES AND THEOREMS
    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.
  5. REAL QUADRATIC FAMILY: A COMPLETE PICTURE
    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.