FMA335F

Temporary

Complex Dynamics in One Variable

12

Third-cycle course

7151 (Centre of Mathematical Sciences / Mathematics)

2025-10-08

Jonas Johansson

English

If sufficient demand

To give a solid ground for the theory of iterations of complex analytic functions. The course is preparatory for research studies in complex dynamics or nearby areas.

Riemann surfaces and covering maps. Normal families and Montel’s theorem. The Fatou set and the Julia set of a function. Properties of these sets. Linearization around periodic points; Koenigs’ and Böttcher’s theorems, as well as Siegel’s theorem on linearization around neutral fixed points. Dynamics near parabolic fixed points. Sullivan’s theorem on the classification of the Fatou set. The holomorphic fixed point theorem. External rays. Hyperbolic and subhyperbolic mappings.

be able to account for the fundamental concepts in the course, such as normal families, the locally uniform topology, Julia sets and Fatou sets, chaotic behavior, local fixed point theory, external rays, hyperbolic mappings, etc.

with his/her own words explain the connections between the concepts of the course (theorems and proofs).

shall be able to solve a selected number of exercises in the book (given in the beginning of the course).

demonstrate mathematical maturity by being able to reason about the mathematical concepts in the course.

Self-study literature review

Lectures and seminars can be given.

Oral exam

Written assignments

Exercises from the book, and an oral exam.

Failed, pass

Function theory (FMAF01) or Complex analysis (advanced level, MATM32).

Complex analysis at advanced level.

Milnor, J.: Dynamics in one complex variable.. Princeton university press.

FMA335F

2025-10-08

Jonas Johansson