Teaching



2012-2013


University of Copenhagen, Mathematics, Master's level: Dynamical Systems
(Modul: 5010-B3-3f13;
Dynamiske Systemer (Dynsys))
Block 3 (Courses and exercises: Feb 5 to March 21, 2013 / Easter vacation March 28-April 1/ Exam April 11 /Reexam June 27)

Lectures: Tue 10-12 and Thu 13-15, Auditorium 7
There will be a session for questions and finishing the exercises on Tuesday March 26 at 10:15 AM in Aud 7.

Exercises Thu 9-12, 1-0-30 (DIKU) -Teaching assistant: Job Kuit

Exam time and room (SIS page) Be there in good time and don't forget to bring your exam number!
PENSUM: All the definitions and theorems required to understand the written exam questions will be in the Brin-Stuck book. There will be no questions on sections 4.2, 4.3, and 5.
February 5 and 7 (week 6):

1. Introduction: Orbit of a discrete-time dynamical system. Statement of the Birkhoff ergodic theorem.
Examples: circle rotations, angle doubling, subshift of finite type. If time allows: Gauss map and applications to number theory.

2. Topological dynamics:
2.1 Topological conjugacy, transitivity, mixing.
2.2 Topological entropy. (Definitions with open covers and for a metric space.)

Worksheet 1
February 12 and 14 (week 7):

2.2 (Continuation) Topological entropy. (Definitions with open covers and for a metric space.)

3. Ergodic theory.
3.1. Invariant measures. Krylov-Bogolyubov theorem.
3.2. Poincaré recurrence theorem
3.3. Ergodicity and mixing. Relation with topological transitivity and topological mixing. Birkhoff ergodic theorem (statement).

Worksheet 2
February 19 and 21 (week 8):

3.3. (Continuation.) Birkhoff ergodic theorem (Proof).
3.4. Kolmogorov-Sinai entropy. Definition. Shannon-Mc Millan Breiman theorem (Statement only). Kolmogorov-Sinai theorem about generating partitions (Statement only). Variational principle (Statement)

Worksheet 3
February 26 and 28 (week 9):

3.4. (Continuation) Proof of the variational principle.

4. Differentiable dynamics.
Introduction. Acims and physical measures.
4.1. Differentiable and locally expanding endomorphisms: Definition. The transfer operator. Fixed points of the transfer operator and its dual. Existence of a unique acim (statement of theorem and beginning of proof).

Worksheet 4
First assignment (due date March 5 at 10:15) Answers to first assignment
March 5 and 7 (week 10):

4.1 (Continuation) End of the proof of existence of a unique acim.
Perron-Frobenius theorem (spectral gap) for locally expanding endomorphisms (statement, beginning of proof).

Worksheet 5
March 12 and 14 (week 11):

4.1 (Continuation) End of the proof of the Perron-Frobenius theorem for locally expanding endomorphisms. Corollary on exponential decay of correlations.
4.2. Hyperbolic diffeomorphisms: Hartman-Grobman linearisation theorem (statement and proof).

Worksheet 6
Second assignment (due date March 22 at 14:00, in office 04.1.03 (Nina and Lone)) Answers to second assignment
March 19 and 21 (week 12):

4.2. Hyperbolic diffeomorphisms (continuation): Structural stability
4.3. Stable and unstable manifolds

5. Oseledec theorem and Lyapunov exponents (no proofs)

Worksheet 7

April 11 2013 Exam
Answers to the exam

Viviane Baladi