Course description

• Course content
• Learning outcome
• Admission
• Prerequisites

• Overlapping courses
• Teaching
• Examination
• Evaluation

Course content

The course is an introduction to the classical topic of one variable complex dynamics. We study iterations of a map from the Riemann sphere to itself, which we divide into a chaotic part (the Julia set) and an orderly part (the Fatou set). In the first part of the course we use mostly basic complex analysis, and in the second part we use potential theory. This allows us to describe statistical properties of the dynamical system.

Learning outcome

After completing the course you will have knowledge of the basic theory about the following, and be able to use the knowledge to solve problems:

• complex dynamics on the Riemann sphere;
• local dynamics: fixed points and periodic points;
• global dynamics: Fatou- and Julia sets, self similarity, classification of periodic Fatou components, Sullivan¡¯s non-wandering theorem;
• topological entropy and invariant measures;
• equilibrium measure: potentials and quasi-potentials, construction of the equilibrium measure, equidistribution, mixing, Lyapounov exponents.

Admission

PhD candidates from the University of Oslo should apply for classes and register for examinations through Studentweb.

If a course has limited intake capacity, priority will be given to PhD candidates who follow an individual education plan where this particular course is included. Some national researchers¡¯ schools may have specific rules for ranking applicants for courses with limited intake capacity.

PhD candidates who have been admitted to another higher education institution must apply for a position as a visiting student within a given deadline.

Prerequisites

Recommended previous knowledge

MAT2400 ¨C Real Analysis and MAT2410 ¨C Introduction to Complex Analysis. It will be useful to have MAT4800 ¨C Complex Analysis and some knowledge of measure theory, but it's not absolutely necessary.

Overlapping courses

10 credits overlap with MAT4820 ¨C Complex Dynamics

*The information about overlaps for discontinued courses may not be complete. If you have questions, please contact the Department. 

Teaching

4 hours of lectures/exercises per week throughout the semester.

Upon the attendance of three or fewer students, the lecturer may, in conjunction with the Head of Teaching, change the course to self-study with supervision.

Examination

1 mandatory assignment.

Final oral examination, including oral presentation of a topic of own choice and following examination.

Examination support material

No examination support material is allowed.

Language of examination

Subjects taught in English will only offer the exam paper in English.

You may write your examination paper in Norwegian, Swedish, Danish or English.

Grading scale

Grades are awarded on a pass/fail scale. Read more about the grading system.

Explanations and appeals

• Explanation of grades and appeals

Resit an examination

Students who can document a valid reason for absence from the regular examination are offered a postponed examination at the beginning of the next semester.

Re-scheduled examinations are not offered to students who withdraw during, or did not pass the original examination.

Withdrawal from an examination

It is possible to take the exam up to 3 times. If you withdraw from the exam after the deadline or during the exam, this will be counted as an examination attempt.

Special examination arrangements

Application form, deadline and requirements for special examination arrangements.

Evaluation

The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.