Help outside of classes
- Office hours: Tuesday and Thursday 9:40-11:00 AM or
by appointment.
Handouts
Outline of the course.
- Basics of Holomorphic Dynamics.
(Riemann surfaces. Schwarz lemma. Universal cover. Poincare metric.
Normal families. Iterations of holomophic maps, Julia set. Local theory
of fixed points. Hyperbolic and sub-hyperbolic maps.)
- Self-similar groups and iterated
monodromy groups.
(Self-similar groups. Iterated monodromy groups. Contracting groups and
their limit spaces. Limit spaces of iterated monodromy groups.
Examples.)
- Quadratic family.
(Mandelbrot set. Iterated monodromy groups of quadratic polynomials.
Kneading sequences and symbolic dynamics of quadratic polynomials.
Mating and examples of plane-filling curves.)
Reading:
Dynamics
in one complex variable, by
J. Milnor, Third edition. Annals of Mathematics Studies, 160. Princeton
University Press, Princeton, NJ, 2006.
This book will be used very much in the first part of the course
(Basics of Holomorphic Dynamics). A draft version of this book
can be
found on Arxiv.
Self-similar
groups, by V. Nekrashevych, published by American
Mathematical Society, ``Mathematical Surveys and Monographs'' vol. 117.
This book will be used in second (Self-similar groups and iterated
monodromy groups) and partly in third (Quadratic family) parts. You can
also use the following preprints.
Self-similar
groups and their geometry
Symbolic
dynamics and self-similar groups
Advanced and additional reading (more links will be added later):
- From fractal
groups to fractal sets (by L.Bartholdi, R.Grigorchuk and
V.N.)
- Iteration of rational
functions. Complex
analytic dynamical systems, by Alan F Beardon,
Graduate Texts in Mathematics, 132. Springer-Verlag, New
York, 1991
- Periodic
orbits, externals rays and the Mandelbrot set: an expository account,
by J. Milnor, Géométrie complexe et systèmes
dynamiques (Orsay, 1995). Astérisque No. 261 (2000),
xiii, 277--333. (A preliminary version is available on Arxiv).
- Symbolic Dynamics of Quadratic Polynomials,
by Henk Bruin and Dierk Schleicher