MATH43320: Ergodic Theory

• Prior knowledge of Analysis

• None

• None

• 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.

• 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.

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.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• 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.

Weekly written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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