This course provides an introduction to ergodic theory. This subject uses mathematical analysis to explore the statistical properties of deterministic dynamical systems, such as their long run average behaviour. In particular, the course will explore applications and extensions of the famous Poincare Recurrence Theorem, and Birkhoff Ergodic Theorem for a measure preserving systems. Throughout the course, fundamental concepts will be explored using various dynamical system case studies, of relevance to both pure topics (such as number theory) and applied disciplines (in particular, the characterization of deterministic chaotic behaviour in nonlinear dynamical systems).

By taking this module, you will gain a deeper understanding and mathematical analysis of dynamical systems, and the statistical and recurrence properties of their orbits. This is especially in the case of chaotic dynamical behaviour.

Module Specific Skills and Knowledge:
1 Recall and apply key definitions and theoretical results within ergodic theory, and dynamical systems.
2 Apply concepts in ergodic theory to dynamical system case studies. Use mathematical techniques required for proving theorems, and for understanding recurrence properties of dynamical systems.

Discipline Specific Skills and Knowledge:
3 Extract abstract mathematical formulations from a diverse range of problems.
4 Apply abstract reasoning and rigorous analysis is to solve a large range of problems.

Personal and Key Transferable/ Employment Skills and Knowledge:
5 Show ability to think analytically and to use rigorous arguments to formulate solutions as mathematical proofs.
6 Communicate results in a clear, correct and coherent manner.
 

-- Overview of relevant probability and measure theory.

-- Basic dynamical systems theory and motivating case studies.  

-- Invariant measures for dynamical systems and Poincare' Recurrence.

-- Ergodicity, mixing and the Birkhoff Ergodic Theorem.

-- Dynamical system case studies: exploring ergodicity and recurrence statistics for selected examples. Examples include irrational rotations, Markov maps, uniformly expanding maps, subshifts of finite type, and the Gauss map. 

The course will also explore a subset of the following special topics:

i) Ergodicity of non-uniformly expanding maps. Examples include chaotic dynamical systems such as the quadratic map, the Lorenz family of maps, and maps displaying intermittency.

ii) Recurrence properties of hyperbolic systems, e.g. the Arnold Cat Map, the Solenoid Map.

iii) Connections with probability theory for independent identically distributed random variables, e.g. quantifying deviations from the average via the Central Limit Theorem, and quantifying infinitely often recurrence via Borel-Cantelli theory.

iv) Connections between the statistics of recurrence and extreme value theory.

v) Connections with number theory (e.g. Weyl's theorem on polynomials).

vi) Symbolic dynamics and Gibbs measures.

vii) Thermodynamic formalism: topological and metric entropy, topological pressure.

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped. 

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%.

Web based and electronic resources:

ELE – College to provide hyperlink to appropriate pages

Reading list for this module: