Dynamical systems
Spring semester 2021 in Fribourg

Lecture: Monday 10h-12h, room 2.73 of the Physics building PER08.
Lecture and Exercises: Tuesdays 10h-12h, room 2.52 of the Physics building PER08.

Content

The aim of this course is to give an introduction to the theory of dynamical systems. The main goal is to get an overview over the basic general notions and examples, linear systems, non-negative matrices and their applications, and ergodic theory. We will, however, have a particular focus on certain examples of low-dimensional dynamical systems: homeomorphisms of the circle, vector fields in the plane and homeomorphisms of surfaces.

Program

The program below is updated weekly and shows our progress. See also the course's Moodle page. There are lecture notes for the course. The notes will be updated regularly during the semester. The exercises contained in the lecture notes are discussed on Tuesdays, 11h-12h.

Lecture notes (last update: 31.05.2021, this version contains all chapters)



Date	Lecture topics
February 22 and 23	Introduction, history, first notions and examples of discrete-time dynamical systems. Chapters 0 and 1.1.
March 1 and 2	First notions and examples of continuous-time dynamical systems. Chapter 1.2.
March 8 and 9	Discrete time vs. continuous time, and linear dynamical systems. Chapters 1.3 and 2.1-2.2.
March 15 and 16	Conjugacy to the linear part, the Hartman-Grobman theorem. Chapter 2.3.
March 22 and 23	Nonnegative matrices, Perron-Frobenius theory. Chapter 2.4.
March 29 and 30	Topological dynamics. Chapter 3.
April 12 and 13	Vector fields in the plane. Chapter 4.
April 19 and 20	Proof of Poincaré-Bendixson, homeomorphisms of the circle. Chapters 4.4-5.1.
April 26 and 27	Rational rotation numbers, Denjoy examples. Chapters 5.1-5.3.1.
May 3 and 4	Irrational rotation numbers, diffeomorphisms of the circle. Chapters 5.3-5.5.
May 10 and 11	Poincaré recurrence, ergodic dynamical systems. Chapters 6.1-6.3.
May 17 and 18	Proof of Birkhoff's Ergodic Theorem, surface topology basics. Chapters 6.3-7.1.
May 25	Thurston's construction of Pseudo-Anosov mapping classes. Chapter 7.2.1.
May 31 and June 1st	Pseudo-Anosov maps, question session. Chapters 7.2.2-end

Literature

The follwing two books are the main sources for the course:

Brin, Stuck "Introduction to dynamical systems"
Katok, Hasselblatt "Introduction to the modern theory of dynamical systems"

Further references are given in the lecture notes.