Complex Dynamics
Lecturer: Alan Sola (SU)
Email: sola (no spam please) math (dot) su (dot) se
Description: Beautiful images of the Mandelbrot set and of Julia sets of polynomial and rational maps became popular in the 1980 and have entered the public imagination as the archetypal fractal sets. The subject of rational iteration is of significantly older date, however, with
a systematic theory arguably having been initiated by Fatou and Julia in the early 20th century.
The aim of this reading course is to
familiarize PhD students with the basic concepts of complex dynamics in one variable, and some of the fundamental theorems concerning the Fatou and Julia sets associated with rational maps.
Please note that this will be a reading course. We will try to have regular meetings to discuss the material but participants will be expected
to read large parts of the book on their own (on in smaller informal reading groups).
Content (to be confirmed):
-Elements of rational iteration: normal families, qc maps, fixed points and multipliers, Fatou sets and Julia sets.
-Fatou components: Basins of attraction, Parabolic sets, Siegel disks. (Herman rings w/o proof)
-Julia set: repelling periodic points, critical points, subhyperbolicity, local connectivity.
-Wandering domains: existence for entire functions (Baker), non-existence for rational maps (Sullivan).
-Mandelbrot set: relation with quadratic polynomials, some elementary features (eg shape of main cardioid).
Textbook:
L. Carleson and T. Gamelin, Complex Dynamics, Universitexts, Springer-Verlag 1993.
(Also: A. Beardon, Iteration of rational functions, GTM 132, Springer-Verlag 1991.)
Examination:
Classroom presentation and oral examination.
Prerequisites: Complex Analysis and Advanced Real Analysis I; Advanced Real Analysis II and Fourier Analysis.
Tentative schedule: Weekly meetings on Tuesdays 3-5pm.
A tentative outline of the course can be found here:
here.
First meeting: Friday September 10 at 10am, room 14, bld 5, SU
(Please contact me via email if you are unable to attend but would like to
participate in the course.)