# Ergodic theory & its applications to number theory

## Summary

This is an introductory course in ergodic theory, providing a comprehensive overlook over the main aspects and applications of this field.

## Content

In the broadest sense, 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, especially number theory and combinatorics. This course provides an introduction to the basics of ergodic theory. Among other things, this includes the theory of recurrence, the structure and convergence of ergodic averages, and the notion of entropy. We will motivate the main ideas and results through simple examples. Another focal point lies on the many groundbreaking applications of ergodic theory in number theory.

## 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:

- Apply tools and techniques from ergodic theory in number theory and combinatorics
- Prove results in ergodic theory
- Formalize dynamcial ideas and concepts such as ergodicity, entropy, chaos, determinism, etc.
- Interpret examples of dynamical systems

## Transversal skills

- Use a work methodology appropriate to the task.
- Continue to work through difficulties or initial failure to find optimal solutions.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking

## Teaching methods

in-person lectures, in-person exercise sessions with the teaching assistant

## Assessment methods

oral exam

## Supervision

Office hours | Yes |

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

- Ergodic Theory with a view towards Number Theory / Einsiedler
- An Introduction to Ergodic Theory / Walters

## Notes/Handbook

Lecture notes will be provided

## Moodle Link

## In the programs

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Ergodic theory & its applications to number theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Ergodic theory & its applications to number theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Ergodic theory & its applications to number theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks