Introduction to dynamical systems

This course is an introduction to subject of dynamical systems from a mathematical point of view. In the first part of the course we will consider various diverse and representative examples. These key examples include,

• circle rotations;
• expanding circle maps;
• shift maps;
• hyperbolic automorphisms of the torus;
• the Gauss map.

Motivated by the examples, we introduce the phenomena and main concepts which one is interested in studying. In the subsequent parts of the course, we further refine and develop these concepts. These topics include the following:

• topological dynamics;
• symbolic dynamics;
• ergodic theory;
• showing ergodicity for hyperbolic systems.

The field of dynamical systems is vast and so we can't be comprehensive in every topic, instead we will work with some of the most significant results and always aim to get an accurate flavour relevant dynamical results and dynamical style arguments.

These pages contain the lecture notes and practical details related to the course.

Practical details

• Second semester of academic year 2023/24
• 8 CFU, 64 lecture hours
• Part of the laurea magistrale in mathematics at the University of Rome Tor Vergata.
• Instructor: Oliver Butterley
• Schedule: see full schedule details.

Further reading

• Introduction to Dynamical Systems by Michael Brin, Garrett Stuck
• A First Course in Dynamics: with a Panorama of Recent Developments by Boris Hasselblatt, Anatole Katok
• An Introduction To Chaotic Dynamical Systems by Robert L. Devaney
• Lecture notes by Carlangelo Liverani
• Lecture notes by Stefano Luzzatto
• ICTP summer school

General advice

• Develop your intuition, it's a powerful skill – But don’t trust it completely
• Don’t aim to memorize but rather seek to understand – It is easy to remember anything when you understand it.
• Question always, be sceptical of all statements presented to you. Don’t accept them until you are sure they are believable.
• Observe, question how everything fits together, notice all the details.
• Part of the process of mathematical reasoning is creative - to be creative we must drop our inhibitions and be ready to be wrong, repeatedly.