# MA9M8 - Topics in Ergodic Theory

**Lecturer: Richard SharpLink opens in a new window**

**Term: **Term 2

**Commitment:** Lectures 30 sessions 1 hour

**Timetable:** Monday 11:00-13:00, D1.07, Thursday 14:00-15:00 D1.07

**Assessment:** Oral Examination 100%

**Prerequisites:**

Basic measure theory and functional analysis.

**Content:**

The module is aimed at beginning PhD students in this area and other graduate students who wish to broaden their knowledge. It does not assume prior knowledge of the subject. The first half will cover basic concepts and we will then go on to introduce several more advanced topics which lead towards current research.

Outline syllabus:

Basic concepts: invariant measures, ergodicity, mixing, unique ergodicity, von Neumann ergodic theorem, Birkhoff ergodic theorem.

Examples of ergodic systems and standard constructions. Existence of ergodic measures for continuous transformations.

Entropy: measure-theoretic entropy, relation to topological entropy, calculation in examples.

More advanced topics may include some of there following:

Precise statistical properties: central limit theorem, rates of mixing, approximation by Brownian motion.

Thermodynamic formalism: pressure as a weighted entropy, variational principle, equilibrium states.

Distribution of orbits of hyperbolic systems: probabilistic and analytic techniques, transfer operators, zeta functions, applications to geometry.

Infinite ergodic theory: skew-product extensions, concepts from random walks on groups.

**References**

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer-Verlag, 1982.

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics 151, Cambridge University Press, 2016.