# MA950 - Circle Diffeomorphisms

**Lecturer: **Dr Selim Ghazouani

**Term(s):** Term 2

**Commitment:** 30 lectures

**Assessment:** Oral exam

**Prerequisites: **Familiarity with topics covered in MA424 Dynamical systems

**Content:**

The circle is arguably the simplest closed manifold, it is therefore a very natural starting point to develop the theory of smooth dynamical systems. The lectures will cover various aspects of the dynamics of circle diffeomorphisms which are ubiquitous in dynamical systems: topological dynamics, ergodic theory, typical behaviour, geometric properties, the importance of regularity.

**I-Topological and ergodic properties of circle diffeomorphisms**

*a) Examples*

*b) Rotation number*

*c) Ergodic properties of circle homeomorphisms*

*d) Topological classification*

*e) Parameter families*

*f) Denjoy theory*

*g) Denjoy counterexample*

*II- Geometric theory*

*a) Arithmetic of the rotation number*

*b) The smooth conjugacy problem*

*c) Small divisors and the cohomological equation*

*d) KAM method and a local conjugacy result*

**References: **

- *An introduction to the modern theory of dynamical systems* (Katok&Hasselblatt)

- *Small denominators I : mapping the circumference onto itself* (Arnold)

- *Sur les courbes définies par les équations différentielles à la surface du tore *(Denjoy)