DYNAMICAL SYSTEMS
| Title | ΔΥΝΑΜΙΚΑ ΣΥΣΤΗΜΑΤΑ / DYNAMICAL SYSTEMS |
| Code | Θ016 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Andreas Koutsogiannis |
| Common | Yes |
| Status | Active |
| Course ID | 600025905 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2025-2030)
Registered students: 0
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Elective Courses belonging to the selected specialization | Winter | - | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Winter |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600268382
|
Course Type 2021
Specific Foundation
Mode of Delivery
- Face to face
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600268382
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
- Greek (Instruction, Examination)
- English (Instruction, Examination)
Learning Outcomes
Upon successful completion of the course, students will
• know the basic features of Dynamical Systems Theory through the study of orbits (ergodicity, minimality)
• understand the basic theorems, concepts and techniques in the study of dynamical systems (theorems of Birkhoff and Furstenberg-Weiss)
• understand the connection between the theory of topological dynamical systems with combinatorics (relation between van der Waerden's theorem and that of Furstenberg-Weiss, as well as its multidimansional version with Gallai's theorem)
General Competences
- Apply knowledge in practice
- Retrieve, analyse and synthesise data and information, with the use of necessary technologies
- Work autonomously
- Advance free, creative and causative thinking
Course Content (Syllabus)
Goal of the course is the study of the following topics:
• Topological Dynamical Systems
• (Point, Set) Recurrence
• Uniform recurrence and minimality
• Existence of periodic orbits
• Non-wandering sets
• Circle rotations
• Shift and subshift operators
• Gauss transformation
• Examples (Horceshoe, Solinoid, Tent function)
• Combinatorial applications
• Birkhoff's multiple recurrence theorem
• Multidimensional van der Waerden theorem
Keywords
Orbit, Recurrence, Ergodicity, Minimality
Educational Material Types
- Notes
- Book
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 39 | ✓ | ✓ | ||
| Reading Assigment | 258 | ✓ | ✓ | ||
| Exams | 3 | ✓ | ✓ | ||
| Total | 300 |
Student Assessment
Description
Exercises: During the talks various exercises are assigned. The students must submit them by the end of the course. These are counted positively towards the final grade. In particular, usually the students take up to 2/10 points from the exercises.
Exam: There is one final written exam at the end.
Final grade: The final grade is a combination of the exercises' grade and the final written exam.
Student Assessment methods
- Written Exam with Multiple Choice Questions (Formative, Summative)
- Written Exam with Short Answer Questions (Formative, Summative)
- Written Exam with Extended Answer Questions (Formative, Summative)
- Written Assignment (Formative, Summative)
- Performance / Staging (Formative, Summative)
- Written Exam with Problem Solving (Formative, Summative)
Bibliography
Additional bibliography for study
• M. Brin and G. Stuck. Introduction to Dynamical Systems, Cambridge University Press, 2004.
• H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981.
• B. Hasselblatt and A. Katok. A First Course in Dynamics with a panorama of recent developments, Cambridge University Press, 2003.
• J. de Vries. Topological Dynamical Systems, An Introduction to the Dynamics of Continuous Maps, De Gruyter, 2014.
• P. Walters. An Introduction to Ergodic Theory, Springer-Verlag, 1982.
Last Update
03-12-2024
