Content: Dynamical Systems is one of the most active areas of modern mathematics. This course will be a broad introduction to the subject and will attempt to give some of the flavour of this important area.

The course will have two main themes. Firstly, to understand the behaviour of particular classes of transformations. We begin with the study of one dimensional maps: circle homeomorphisms and expanding maps on an interval. These exhibit some of the features of more general maps studied later in the course. Whilst studying circle maps, we will look at Sharkovsky's Theorem, a beautiful result that describes the periodic orbit structure of circle maps. A second theme is to understand general features shared by different systems. We will see many examples of dynamical systems in different settings, including the complex plane, tori and more. We will also see how to use techniques involving dynamical systems to tackle problems in number theory.

Aims: We will cover some of the following topics:

• circle homeomorphisms and minimal homeomorphisms
• Sharkovsky's Theorem
• expanding maps and Julia sets
• symbolic dynamics
• applications of dynamical systems to problems in number theory
• examples of complex dynamical systems

Books:

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press

B. Hasselblatt and A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge University Press

S. Sternberg, Dynamical Systems, Dover

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag