Duration
30h Th, 20h Pr
Number of credits
| Master in mathematics (120 ECTS) (Even years, not organized in 2023-2024) | 8 crédits |
Lecturer
Language(s) of instruction
French language
Organisation and examination
Teaching in the first semester, review in January
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
Ergodic theory is a field of mathematics that studies the evolution of a dynamical system over time. If X is a set and T:X->X an application, we will be interested in the behavior of T^n(x) for x in X, when n varies. In this course, we will study two types of such systems: those called "topological" (X is a compact metric space and T is a continuous application) and those called "measured" (X is a probability space and T is a measurable and measure-preserving application). An example of a classical result is Birkhof's ergodic theorem which makes a link between "spatial" measure of a measurable set A and "temporal" measure: under certain conditions, almost all points of the system visit A (under the action of T) with a frequency equal to the measure of A. In particular, we will be interested in applying the results encountered in number theory.
Learning outcomes of the learning unit
The student will master fundamental notions seen during the lectures as well as the corresponding proofs. He will be able to present them clearly and succinctly. Also, he will be able to apply those notions in order to solve related problems.
Prerequisite knowledge and skills
Basic mathematical knowledge from a bachelor degree in mathematics.
Knowledge of topology and measure theory is a plus, but we will review the needed notions.
Planned learning activities and teaching methods
Theoretical lectures and exercise sessions using "blackboard and chalk" or beamer, interacting with students.
Mode of delivery (face to face, distance learning, hybrid learning)
Lectures are mainly dedicated to theoretical aspects. Pratical sessions are devoted to solve exercises and to enlighten the concepts presented during the lecture.
The course and exercise sessions will be given face-to-face, but depending on the evolution of the health situation, could also be given online.
Depending on the number of students, the course might consists in a lecture group in which students will be asked to prepare and present some parts of the course.
Recommended or required readings
Reference books:
- M. Einsiedlr, T. Ward. Ergodic theory - with a view toward number theroy. Springer (2011)
- P. Walters. An introduction to ergodic theory. Springer (1982)
- K. Dajani, C. Kraaikamp. Ergodic theory of numbers. The Mathematical Association of America (2002)
An oral examination devoted to the theory (mainly statements and proofs of theorems and discussion) but also direct applications of the theory is organized. Homeworks could possibly be taken into account for the final grade.
In case of a reading group, the personal work and the presentations of the students will be part of the note.
Work placement(s)
Organisational remarks and main changes to the course
Contacts
J. Leroy, Institut de Mathématique (B37) - Allée de la découverte 12 - Sart Tilman, 4000 Liège Tél. : (04) 366.94.70 - E-mail : j.leroy@uliege.be ;