Ergodic theory
Summary
This is an introductory course in ergodic theory, providing a comprehensive overlook over the main aspects and applications of this field.
Content
Ergodic theory is the study of group actions on measure spaces. Its history traces from Poincare's recurrence theorem in celestial mechanics and Boltzman's ergodic hypothesis in statistical physics to its mathematical proliferation in the 1930s through the ergodic theorems of von Neumann, Birkhoff, and Koopman. It has since grown into a hugely important research area with striking applications to other areas of mathematics. This course provides an introduction to the basics of ergodic theory. This includes the structure and convergence of ergodic averages, the theory of recurrence, and the notion of entropy. We will motivate the main ideas and results through simple examples.
Keywords
ergodic theory, dynamcial systems, measure-preserving transformation, entropy
Learning Prerequisites
Recommended courses
Measure and integration
Important concepts to start the course
This course is aimed at master's or advanced bachelor's students. Since ergodic theory is largely based on the notions of measure theory, either some background in measure theory or the willingness to learn some of this material along the way is expected. I will provide a handout summarizing the prerequisites from measure theory that are needed for this course at the beginning of the semester.
Learning Outcomes
By the end of the course, the student must be able to:
- Formalize dynamcial ideas and concepts such as ergodicity, entropy, chaos, determinism, etc.
- Apply tools and techniques from ergodic theory in other areas
- Interpret examples of dynamical systems
- Prove results in ergodic theory
Transversal skills
- Use a work methodology appropriate to the task.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
- Continue to work through difficulties or initial failure to find optimal solutions.
Teaching methods
in-person lectures, in-person exercise sessions with the teaching assistant
Assessment methods
oral exam
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | Yes |
Resources
Bibliography
- M. Einsiedler, T. Ward, Ergodic Theory with a view towards Number Theory, Springer-Verlag London, 2011.
- P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer New York, 1982.
Ressources en bibliothèque
- An Introduction to Ergodic Theory / Walters
- Ergodic Theory with a view towards Number Theory / Einsiedler
Notes/Handbook
Lecture notes will be provided
Moodle Link
In the programs
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Ergodic theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Ergodic theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Ergodic theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Ergodic theory
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional