Undergraduate Programme and Module Handbook 2024-2025

# Module MATH4361: Ergodic Theory IV

## Department: Mathematical Sciences

### MATH4361: Ergodic Theory IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Analysis III (MATH3011)

#### Corequisites

- None

#### Excluded Combination of Modules

- 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: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 3 hours | 100% | none |

#### Formative Assessment:

Eight assignments to be submitted.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University