Course description: The course aims to give an introduction to two disciplines, dynamical systems and Markov chains, which both provide efficient mathematical tools to describe phenomena evolving in time. While dynamical systems and Markov chains share several common features, they are also somewhat complementary in the sense that they study time evolution from the deterministic and the stochastic perspectives, respectively. Of course, there is a lot more to say about the mathematical aspects of time evolution. Yet, the present course focuses on simple examples – dynamical systems in one or two dimensions, and Markov chains with discrete state spaces – which can be studied by elementary tools. This way the students can learn about the main conceptual aspects and the key phenomena without going too deep into the technical complications. It is expected that the experience gained at studying these simple examples will be utmost useful at later stages of their curriculum. Dynamical systems and Markov chains are essential in several major areas of pure and applied mathematics, extensively used for example in financial mathematics or internet search engines, and occur as models for a wide range of engineering, economical, physical, biological and sociological phenomena. Throughout the course, an emphasis is put on highlighting such connections.

Topics:

Dynamical systems: Phase spaces and maps. Regular and chaotic behavior illustrated in simple examples. Periodic points. Density and equidistribution of orbits. One dimensional maps, the logistic family. Emergence of fractals. Maps of the plane, phase portraits. Hyperbolic dynamics and the shadowing property.

Markov chains: The concept of a Markov chain (finite state space, discrete time). Transition probabilities. Classification of states. Transience and recurrence. Irreducible classes. Stationary distributions. Irreducibility and (a)periodicity. Limit behavior. A glimpse at infinite state space.