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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)