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Undergraduate Handbook 2025-2026 (archived)

Module MATH4361: Ergodic Theory IV

Department: Mathematical Sciences

MATH4361: Ergodic Theory IV

Type	Level	Credits	Availability	Module Cap	Location
Open	4	20	Available in 2025/2026	None.	Durham

Pre-requisites

Analysis III (MATH3011)

Corequisites

None

Excluded Combinations

None

Aims

To introduce key concepts in the pure mathematical description of discrete dynamical systems and to study these concepts in a body of examples. To prove and apply major classical theorems from the field.

Content

Dynamical systems in dimension one: circle rotations; doubling map; expanding maps.
Baker’s map; symbolic dynamics.
Concepts from topological dynamics: minimality; topological conjugacy; topological mixing; topological entropy.
Concepts from ergodic theory: invariant measures; ergodicity; mixing; Markov measures, metric entropy.
Poincaré recurrence; Birkhoff’s Ergodic Theorem; Perron-Frobenius Theorem; The ergodic theorem for Markov Chains; Variational principle.

Learning Outcomes

Subject-specific Knowledge

By the end of the module students will:
Be able to solve novel and/or complex problems in the given topics.
Have a knowledge and understanding of this subject demonstrated through an ability to compute the topological/metric entropy of a variety of important dynamical systems and to be able to tell if these are ergodic, minimal or (topologically) mixing.
Be able to reproduce theoretical mathematics related to this course at a level appropriate for Level 4, including key definitions and theorems.

Subject-specific Skills

Students will have developed advanced technical and scholastic skills in the areas of Ergodic Theory and Dynamical Systems.

Key Skills

Students will have highly specialised skills in the following areas: problem solving, abstract reasoning.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total Hours
Lectures	42	2 per week in Michealmas and Epiphany; 2 in Easter	1 Hour	42
Problems Classes	8	Fortnightly in Michaelmas and Epiphany	1 Hour	8
Preparation and Reading				150
Total				200

Summative Assessment

Component	Component Weighting	Element	Length/Duration	Element Weighting	Resit Opportunity
Examination	100%	Written examination	3 hours	100%

Formative Assessment

Eight assignments to be submitted.

Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.