Dr Davoud Cheraghi

Davoud Cheraghi

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One dimensional real and complex dynamics

TCC, Spring 2014

Course description:

In this course we study the iterations of holomorphic maps on the Riemann sphere. The main goal is to understand "global" orbit "structures" of the dynamics of such maps in various forms. The course has two parts: the first part gives a quick introduction to the foundations of the iterations of rational functions, and the second part is devoted to some advanced topics in the subject to introduce the students to various aspects of research in the field.

In the first part, we start with general Fatou/Julia decomposition according to stability of nearby orbits, and the basic properties of these invariant sets. Then, we continue to discuss the classification of Fatou components, and various ways to describe the dynamics on the Julia sets. While the orbits are chaotic on the Julia sets, one may resort to techniques from ergodic theory to extract some information on long term behaviour of orbits; measure theoretic attractors, and physical measures.

In the second part, we plan to introduce some modern techniques that has been recently used successfully to establish some progress in the development of the field. In particular, we introduce puzzle technique and its application to the study of the problem of Lebesque measure of Julia sets; parabolic bifurcation and dynamics of near-parabolic maps through parabolic renormalization; polynomial-like renormalization of Douady-Hubbard to explain the appearance of little Mandelbrot copies in the Mandelbrot set and connections to some deep phenomenon on rigidity and universality.

Beside originating some very effective techniques in the general theory of dynamical systems, the iterations of rational maps has had many interactions with other areas of mathematics, like quasi-conformal mappings and elliptic PDE's, Teichmuller theory, Diophantine approximations, and group theory. If you have curiosity and appetite to see one of the most beautiful branches of mathematics, this course is for you!

We plan to cover the following topics:

  • Introduction to the course, preliminaries from complex analysis, normal families, Montel's theorem;
  • Basics of complex dynamics, Fatou/Julia sets, periodic cycles and their types;
  • Classification of Fatou components, local and global fixed point theories;
  • Measurable dynamics on the Julia sets, measure theoretic attractors, ergodic theory;
  • Dynamics of polynomials, jigsaw puzzle construction;
  • Renormalization of polynomials, polynomial-like mappings, little Mandelbrot copies in the Mandelbrot set;
  • Parabolic bifurcation and parabolic renormalization;
  • Conformal invariants, distorsion theorems, quasi-conformal mappings, measurable Riemann mapping theorem.

Instructors: Davoud Cheraghi

Course prerequisites: complex analysis, a basic knwoledge of dynamcial systems will be helpful but not necessary.

Schedule: Tuesdays 2-4pm (at Imperial College the lectures will be in room 6M42); the first lecture is on Jan 21, and the last one on March 11.
The course consists of eight lectures.

Lecture notes:
Course Notes

Homeworks:
There are some exercises in the lecture notes to help you understand the material. There is no need to hand in your solutions.

Course Assessment: If you are taking the course for credit, you may write a mini project on a topic close to the material covered in the course, at a level accessible to others participating in the course. Mini projects may be tailored to individual student's interests.

References:

Below is a rather wide list of relevant literature on one dimensional real and complex dynamics.
We plan to touch upon appropriate parts of the materials 5, 6, 7, 8, 9, 10, 11 from the list below.

Books and relevant papers on real and complex dynamics:
  1. A. Beardon, The Iteration of Rational Functions: Complex Analytic Dynamical Systems, Springer-Verlag, 1991.
  2. P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984) 85-141.
  3. L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, 1993.
  4. W. de Melo, S. van Strien One dimensional dynamics, Springer-Verlag, 1993.
  5. A Douady, J.H. Hubbard Orsay notes, 1982.
  6. A Douady, J.H. Hubbard On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. (4) 18 (1985), no. 2, 287-343.
  7. M. Lyubich, On the Lebesgue measure of the Julia set of a quadratic polynomial, Preprint 1991.
  8. C. McMullen, Complex Dynamics and Renormalization, Annals of Math Studies 135, Princeton University Press, 1996.
  9. J. Milnor, Dynamics in One Complex Variable, Third edition. Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006.
  10. J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Asterisque No. 261 (2000).
  11. S. Morosawa,Y. Nishimura, M. Taniguchi, Holomorphic dynamics, Cambridge Studies in Adv. Math., 66. Cambridge University Press, Cambridge, 2000.
  12. M. Shishikura, Bifurcation of parabolic fixed points, (chapter in the book) The Mandelbrot set, theme and variations, 325-363, Cambridge Univ. Press, 2000.
  13. N. Steinmetz, Rational Iteration : Complex Analytic Dynamical Systems, Walter de Gruyter, 1993.

Some survey articles on real and complex dynamics: Background in complex analysis and quasi-conformal mappings:
  • L. V. Ahlfors, Complex analysis, Int. Series in Pure and Applied Math., McGraw-Hill Book Co., 1978.
  • L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill, 1973.
  • L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, 1966.
  • J. B. Conway, Functions of one complex variable. Second edition. Graduate Texts in Mathematics, 11. Springer-Verlag, New York-Berlin, 1978.
  • O. Lehto and K. Virtanen, Quasiconformal mappings in the plane, 2nd edition, Springer-Verlag, 1973.
  • C. Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbucher, Vandenhoeck & Ruprecht, Gottingen, 1975.