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MATH4361: Ergodic Theory IV

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Type Open
Level 4
Credits 20
Availability Available in 2024/2025
Module Cap None.
Location Durham
Department Mathematical Sciences

Prerequisites

  • Analysis III (MATH3011)

Corequisites

  • None

Excluded Combinations 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.
  • Bakers 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.
  • Poincar recurrence; Birkhoffs 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

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week in Michealmas and Epiphany; 2 in Easter1 Hour42 
Problems Classes8Fortnightly in Michaelmas and Epiphany1 Hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
Written examination3 hours100none

Formative Assessment

Eight assignments to be submitted.

More information

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