Universality in one-dimensional maps

Topics in differential equation, APMA 2210, Fall 2007

Overview: The discovery of quantitative universality in 1-D maps by Coullet-Tresser and Feigenbaum was one of the most striking developments in dynamical systems in the 1970s. Despite the explosion of activity since then, many basic questions remain open. The goal of this class is to provide a self-contained and rigorous study of the renormalization theory that explains this universality. We shall aim for a streamlined picture of scattered results. There will only be time to investigate a `classical age' of the theory- from the numerical discoveries of the 1970s, to the introduction of complex bounds in the 1980s.

Topics:
  1. Basics of one-dimensional maps- numerics, heuristics and combinatorics (circle maps, kneading theory, Sharkovskii's theorem).
  2. Complex bounds - univalent functions, Koebe estimates, Beltrami's equation, Fatou and Julia sets, the Pick class and Loewner's theorem.
  3. Renormalization- heuristics, analysis of the Feigenbaum-Cvitanovi\'c equations, thermodynamic formalism, Sullivan's theorems.
Meeting time: 2-2:50 pm, MWF, Room 155, Barus and Holley.

Prerequisites: The formal prerequisites are a semester each of real and complex analysis. What is really required is some mathematical maturity.

Homework: Homework will be posted on the website, typically every two weeks. Grades will be based on homework.

HW 1. Solutions.
HW 2. Solutions. Extra.
HW 3.


Lecture notes: Lecture notes will be developed and posted on the website.

Week 1.
Weeks 2-4. (revised)

The principal sources for these notes are the following, especially the book of de Melo and van Strien. This book is unavailable and out of print, so please do not recall it from the library!


Sources:
  1. Iterated maps on the interval as dynamical systems , P. Collet and J.-P. Eckmann , Birkhauser, 1980. One of the first accounts of the basic dynamical systems picture.
  2. Universality in chaos , edited by P. Cvitanovi\'c, IOP Publishing, Britol, 1984. This is a diverse collection of reprints of classic articles. It includes several papers we will read (especially, the original papers of Feigenbaum).
  3. On iterated maps of the interval , J. Milnor and W. Thurston , Lecture Notes in Mathematics, 1342, Springer, 1988. This is a wonderful article that develops the combinatorial theory of one-dimensional maps.
  4. Bounds, quadratic differentials, and renormalization conjectures , D. Sullivan. Mathematics into the Twenty-First Century, Vol 2, AMS, 1992. Online at http://www.ams.org/online bks/hmbrowder/. A good survey of the achievements of the 1980s that underlies most consequent work.
  5. One-dimensional dynamics W. de Melo, S. van Strien , Springer, 1992. This is the authoritative source on the theory.
  6. Complex dynamics , L. Carleson, T.W. Gamelin , Springer, 1993. This is an introduction to complex dynamics for analysts. It includes all the complex analysis we need (and a lot more).
  7. Complex dynamics and renormalization , C. McMullen, Princeton University Press, 1992. We won't get this far, but its great browsing.


File translated from TEX by TTH, version 3.66.
On 4 Sep 2007, 11:33.