# MA627 Ergodic Theory

**Lecturer: Tim Austin**

**Term(s):** Term 2

**Commitment:** 30 Lectures

**Assessment:** Oral Exam.

**Formal registration prerequisites: **None

**Assumed knowledge:**

- Eigenvalues and eigenvectors

MA222 Metric Spaces / MA260 Norms, Metrics and Topologies:

- Metric spaces
- Continuity
- Compactness

- Abstract measures
- Lebesgue measure
- Convergence Theorems
- L^1 and L^2 spaces

**Useful background:**

ST111 Probability A and ST112 Probability B:

- Probability spaces
- Notion of random variable
- Law of large numbers

- Hilbert Spaces
- Orthonormal basis
- Dual spaces

- Banach spaces
- Hanh-Banach theorem
- Convex sets

- Fourier series and their properties

**Synergies:**

**Leads To:**

**Content**: Consider the following maps:

- A fixed rotation of a circle through an angle which is an irrational multiple of .
- The map of a circle which doubles angles.

If we choose two points of the circle which are close to each other and repeatedly apply the first map the behaviour of each point closely resembles the behaviour of the other point. On the other hand if we apply the second map repeatedly this is no longer the case - the behaviour of each point can be wildly different. The first example can be described as `deterministic' or `rigid' and the second as `random' or `chaotic'. We shall examine many examples of such maps displaying various degrees of randomness, and one of our aims will be to classify different types of behaviour using measure theoretic techniques. A key result (which we will prove) is the ergodic theorem. This is a basic tool in our analysis. We shall also consider applications to number theory and to Markov chains. For most of the module rigorous proofs will be provided. Occasionally we shall give proofs which depend on references which you will be encouraged to read. The written examination will depend only on module lectures.

**Aims**: To study the long term behaviour of dynamical systems (or iterations of maps) using methods developed in Measure Theory, Linear Analysis and Probability Theory.

**Objectives**: At the end of the module the student is expected to be familiar with the ergodic theorem and its application to the analysis of the dynamical behaviour of a variety of examples.

**Books**: **(Recommended reading):**

Introductory (good for advanced undergraduate study):

- William Parry, Topics in Ergodic Theory, Cambridge University Press 1981
- Omri Sarig, online lecture notes maintained at <ergodicnotes.pdf (weizmann.ac.il)>

More advanced (good for PhD students):

- Peter Walters, An Introduction to Ergodic Theory, Springer 1981
- Karl Peterson, Ergodic Theory, Cambridge University Press 1989
- Manfred Einsiedler and Thomas Ward, Ergodic Theory with a view towards Number Theory, Springer 2011

For comprehensive reference:

- Anatole Katok and Boris Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge 1995