Duration
20h Th, 20h Pr
Number of credits
| Bachelor in mathematics | 4 crédits |
Lecturer
Language(s) of instruction
French language
Organisation and examination
Teaching in the second semester
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
This lecture is an introduction to discrete toological dynamical systems. We will study classical aspects of these systems, such as recurrence, minimality, entropy,... We will then focus on symbolic dynamical systems for which the dynamics is given by the shift map on (bi-)infinite words. We will in particular systems of finite type and sofic systems.
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 knowledges in general topology and graph theory.
Planned learning activities and teaching methods
Theoretical lectures using "blackboard and chalk" or beamer, interacting with students. During exercises sessions, students could face exercises that must be solved.
Mode of delivery (face-to-face ; distance-learning)
Lectures are mainly dedicated to theoretical aspects. Pratical sessions are devoted to solve exercises and to enlighten the concepts presented during the lecture. Detailed schedule will be given at the beginning of the academic year
Recommended or required readings
This lecture is mainly based on the book:
P. Kurka, Topological and symbolic dynamics, Societe Mathematique de France, 2003.
Other references:
- D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press (1995)
- M. Queffélec, Substitution Dynamical Systems - Spectral Analysis (Second Edition), Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg (2010)
- P. Walters, An Introduction to Ergodic Theory, Graduate texts in Mathematics 79, Springer -Verlag New York Heidelberg Berlin (1982)
Assessment methods and criteria
The final examination is divided into an oral part and a written exam. The oral exam is devoted to the theory but also direct applications of the theory (student may be asked to solve a small exercise on the blackboard or on a sheet of paper). The written exam evaluate the comprehension of the material of the exercice sessions.
Work placement(s)
Organizational remarks
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@ulg.ac.be