Introductory description

Ergodic Theory is the study of dynamical systems from a statistical or probabilistic point of view.

Central themes are invariant measures and the asymptotic distribution of orbits. It has applications to many other areas of pure mathematics, notably number theory and geometry.

The module will cover both the basic theory and a number of more advanced topics.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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

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

More advanced topics of which the following is an indicative list (the actual content may vary from year to year);_

Equidistribution result and applications to number theory: irrational rotations, Weyl’s theorem, continued fraction expansions.

Recurrence and applications to combinatorial number theory: van der Waerden’s theorem, Furstenberg’s correspondence principle, Szemeredi’s theorem.

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

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

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

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

Learning outcomes

By the end of the module, students should be able to:

• After successfully completing the module, students will have a rigorous understanding of the basic concepts and techniques used in ergodic theory, and be able to apply this to applications. They will also have a deep understanding of several advanced topics that will provide a basis to beginning the study of current research problems.

Subject specific skills

Students will have developed a familiarity with the concepts of ergodic theory have a rigorous understanding of a tool kit of techniques:

• Use of systems built on rotations to study problems in number theory.
• Transference of problems in combinatorial number theory to dynamics.
• Analytic methods to investigate chaotic systems.
• Use of thermodynamic formalism to identify limiting measures.

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)