Summary:

The course will focus on holomorphic dynamical systems in one complex variable.

We will study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. The theory of iterations of rational maps on the Riemann sphere goes back to the works of Julia and Fatou at the beginning of the 20th century, however, the introduction of computer simulations around 1980s has motivated recent dramatic progress in the area.

A large part of the course will focus on the "classical" part of the theory that was mostly developed before 1980s. We will define the Julia and Fatou sets of a rational map (and, more generally, of a holomorphic map of a Riemann surface to itself) and discuss their basic properties. Roughly speaking, the Julia set is the closed invariant set on which the dynamics is chaotic, while the Fatou set is its complement. Then we will study local dynamics of a holomorphic map around a fixed point. This will be a fundamental tool in understanding more global dynamics. We will conclude the classical part by describing the possible structures of the Fatou set. If time permits, in the end of the course we will very briefly touch upon some more recent developments, such as the methods of quasiconformal surgery or the theory of renormalization in holomorphic dynamics.

Main textbook:

John Milnor, Dynamics in One Complex Variable, third ed., 2006

(earlier version: https://arxiv.org/pdf/math/9201272.pdf )

Class Schedule

Tu, 04/02/2020: Introduction. Pictures. Definition of Julia and Fatou sets via Lyapunov stability.

Th, 06/02/2020: Basic properties of Julia and Fatou sets. Multiplier of a periodic orbit (cycle). Attracting, repelling and neutral cycles. Basin of attraction.

Tu, 11/02/2020: Parabolic periodic points. Classification of Riemann surfaces. Exercises 1

Th, 13/02/2020: Classification of Riemann surfaces. Picard's theorem.

Tu, 18/02/2020: The Poincare metric on hyperbolic Riemann surfaces. Holomorphic maps contract the Poincare metric. Exercises 2

Th, 20/02/2020: Normal families. Montel's theorem.

Tu, 25/02/2020: Discussion of homework exercises

Th, 27/02/2020: Montel's theorem. Definition of Julia and Fatou sets via normal families.

Tu, 03/03/2020: Further basic properties of Julia sets of a rational map.

Th, 05/03/2020: Further basic properties of Julia sets of a rational map. Exercises 3

Tu, 10/03/2020: Hausdorff dimension. Smooth Julia sets.

Th, 12/03/2020: Smooth Julia sets. Lattes maps.

Tu, 17/03/2020: Discussion of homework exercises

Th, 19/03/2020: Discussion of homework exercises (continuation)

Tu, 24/03/2020: Formal normal forms. Poincare-Dulac theorem.

Th, 26/03/2020: Holomorphic normal forms. Poincare theorem.

Tu, 31/03/2020: Fatou-Bieberbach domains. Critical points in basins of attraction of rational maps.

Th, 02/04/2020: Bottcher's theorem. Applications to polynomials. Exercises 4

---SPRING BREAK---

Tu, 14/04/2020: Proof of Bottcher's theorem. Short discussion of local connectivity and Caratheodory's theorem.

Th, 16/04/2020: Parabolic fixed points.

Tu, 21/04/2020: Discussion of homework exercises

Th, 23/04/2020: Discussion of homework exercises (continuation). Parabolic fixed points.

Tu, 28/04/2020: Parabolic fixed points.

Th, 30/04/2020: Irrationally indifferent fixed points.

Tu, 05/05/2020: Irrationally indifferent fixed points. Most periodic orbits repel.

Th, 07/05/2020: Most periodic orbits repel. Density of repelling periodic points in J.

Tu, 12/05/2020:

Th, 14/05/2020:

Final Exam: TBA