Course Description: A dynamical system can be obtained by iterating a function or considering evolutions of physical systems in time. Even if the rules of evolution are deterministic, the long term behaviour of the systems are often unpredictable and chaotic. The Theory of Dynamical Systems provides tools to analyse this chaotic behaviour and estimate it on average. It is an exciting and active field of mathematics that has connections with Analysis, Geometry, and Number Theory.

At the beginning of the course we concentrate on presenting many fundamental examples of dynamical systems (such as Circle Rotations, the Baker Map, the Continued Fraction Map, and others). Motivated by theses examples, we introduce some of the important notions that one is interested in studying. Then in the second part of the course we will formalise these concepts and cover the basic definitions and some of the fundamental results in Topological Dynamics, Symbolic Dynamics, and Ergodic Theory. During the course we also discuss applications to other areas of mathematics and to concrete problems such as, for instance, Internet search engines.

References:

• L. Barreira and C. Valls, Dynamical Systems: An Introduction, Springer, 2012.
• M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2015.
• B. Hasselblatt and A. Katok, A First Course in Dynamics: with a Panorama of Recent Developments, Cambdirge University Press, 2003.
• M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, Cambridge University Press, 1998.