Teacher: Kari Astala
Scope: 10 cr
Registration: via WebOodi
Note: Course seminar will be on Wednesday, December 13 at 10 o'clock. Room M2 (M233).
Lauri Hitruhin gives advice and tutorials for seminar presentations on Fridays 12 - 14; room M203.
You can reach him also by email [email: email@example.com ]On Tuesday Dec. 12, Kari Astala is available in his office at 10 -11.
Teaching: Lectures are on Mondays and Tuesdays at 12-14.
NOTE: During period II Monday's lecture is in room M203. Tuesday lecture in room M2 (M233).
First lecture on period II is on Monday 30.10.2017.
For second period grade is obtained (by Exercise 7 and) a seminar presentation.
Last exercise (no. 7) on Friday 10.11.; after that the Fridays 12-14 (room M203) are used for tutorials and guidance for the seminar presentation.
Here are suggestions for a topic; topic of your own is also well-come !
Topics of presentation will be agreed on Fri. 3.11/Exercise class or after the lectures 6.11 or 7.11; you may also contact Kari or Lauri on this directly by email.
Seventh homework here and in the materials folder (to be returned by Fri. 10.11)
Sixth homework here and in the materials folder (to be returned by Fri. 3.11.)
Fifth homework here and in the material folder (to be returned by Fri. 20.10)
Fourth homework here and in the materials folder (to be returned by Fri. 13.10)
Third homework here and in the materials folder (to be returned by Fri. 6.10)
Second homework here and in the materials folder (to be returned by Fri. 29.9)
First homework here and in the materials folder (to be returned by Fri. 22.9).
Homework should be returned to Lauri Hitruhin, either at the exercise class or by email to firstname.lastname@example.org , or to the box at the office door of Kari Astala (room M224)
The course extends over periods I and II; during period II Monday's lecture is in room M203.
Grading: There will be no exam for this course; grading is based on homework and course activity.
For the first period, homework is given weekly, returned in writing and the grade comes from homework (50 % = 1/5; 90 % = 5/5).
For the second period, we will have 1-2 homeworks; grade is obtained by these and a seminar presentation.
Topic: Complex dynamics (i.e. holomorphic dynamics) can be considered as part of two fields, 'Dynamical Systems' and 'Complex Analysis'.
In Complex Dynamics one studies the discrete-time dynamical system, arising from iterating a complex polynomial P(z). That is, setting zn+1 = P(zn), n = 0, 1, 2,... , one wants to understand the asymptotic behaviour of zn, when n → infinity, and see e.g. how this depends on the initial value z0.
Typical questions are for instance the geometric properties of the associated attractor, i.e. the chaotic part of the system (the Julia set, c.f. picture above).
Compared to other dynamical systems, in complex dynamics it is possible to achieve, with tools from complex analysis, a surprisingly detailed understanding of the systems. For complex analysis, complex dynamics allows fascinating applications of the methods and results of the field.
In brief, the aim of the course is to understand these questions mathematically, both local dynamics and, in particular, the geometry of Julia sets and the Mandelbrot set (picture below).
One can study also dynamics under a rational function R(z) or an entire function f(z). If time allows some topics of these will also be discussed.
Prerequisites: The starting point is knowledge of basic complex analysis, e.g. as covered in the course MS-C1300 Kompleksianalyysi (Here are notes for that course). A presentation with more details and complete proofs can be found here and in the materials folder; these are lectures by Kari Astala at Univ.Helsinki, 2016 (a 2 period course), but unfortunately these notes are in Finnish.
Further material from complex analysis, in addition to above notes, will be necessary but these will be covered during the course.
The complete set of course notes (lectures up to Dec. 5.) can be found here and in the materials folder.
To Appendix of notes, a proof of the complex analytic implicit function theorem was added
Literature. There are several books on the topics, for instance:
- Beardon, Iteration of Rational Functions, Springer-Verlag
- Carlson-Gamelin, Complex Dynamics, Springer-Verlag
- Berteloot - Mayer, Rudiments de Dynamique Holomorphe, Lectures (In French)
The background material on basic complex analysis can be found also in (each of) the books:
- Brown - Churchill: Complex Variables and Applications, (8th ed.), McGraw-Hill, 2009.
- J.B. Conway: Functions of One Complex Variable I (2nd ed.), Springer, 1978.
- R.E. Greene - S.G. Krantz: Function Theory of One Complex Variable (2nd ed.), AMS, 2002.
- B. Palka: An Introduction to Complex Function Theory, Springer, 1991.
- W.Rudin: Real and Complex Analysis (3rd ed.), McGraw-Hill, 1987.