2024/2025
Holomorphic Dynamics
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Instructors:
Vladlen Timorin
Language:
English
ECTS credits:
3
Contact hours:
30
Course Syllabus
Abstract
The simplest (from the viewpoint of algebra) formulas, such as $f(z)=z^2+c$, can generate intricate self-similar structures when the corresponding maps are regarded as dynamical systems. Dynamical systems theory studies what the \emph{orbits} of $f$, i.e., sequences of the form $z$, $f(z)$, $f(f(z))$, $\dots$, are doing. Typical questions: what a specific orbit looks like (does it converge to a periodic cycle or exhibit a chaotic behavior)? How does the behavior of the orbit depend on the initial point $z$? How does it change as $f$ itself varies? Rapid development of holomorphic dynamics, which started around 1980s, became possible, among other factors, due to emerging of computer graphics. Unexpected pictures motivated new results, which were rigorously proven later. We will discuss some fundamental results and the simplest examples from the area of complex dynamics, mostly following J. Milnor's textbook.
Learning Objectives
- Students will develop intuition and acquire a mathematical toolkit for working with 1D holomorphic dynamical systems.
Course Contents
- Rational dynamics on the Riemann sphere: examples and pictures
- Riemann surfaces and uniformization (an overview of basic notions and results).
- Fatou and Julia sets; their simplest properties
- Local dynamics near fixed points, linearization
- Hyperbolicity in holomorphic dynamics
- Polynomial dynamics: external rays, the role of local connectedness
