Lecture Course I:

Ergodic theory is a powerful set of mathematical tools and ideas used to study the behaviour of dynamical systems in which the behaviour of individual orbits is very complex, while the behaviour of collections of orbits is sufficiently orderly to allow them to be studied using the language of probability or measure theory. This body of work has found diverse applications in dynamical systems, number theory, geometry and many other parts of mathematics. As a branch of mathematics, it is characterised by the great diversity of techniques employed and by a huge diversity of sources of examples and phenomena. This course is intended to equip the students with the language and background for the other two more specialised courses.

SYLLABUS:

• What is a dynamical system? The significance of compactness and finite measures. What kind of questions can be asked and answered about dynamical systems?
• Recurrence, ergodicity and mixing.
• Vitali covering and the ergodic theorems.
• Continued fractions, beta-transformations, group automorphisms, symbolic models.
• Entropy, complexity and structure.

Lecture Notes

BACKGROUND READING:

• The language of measure theory will be needed, and students not familiar with this should read up on the basics.
• The recent book of Einsiedler and Ward contains all that will be covered (and more). Parts of it will be made available to participants.