Conducted in
term:
2024/25
ISCED code: 0541
ECTS credits:
unknown
Language:
Polish
Organized by:
Doctoral School of Exact and Natural Sciences
(in Polish) Dyfeomorfizmy okręgu 7404-MAT-DYFEO
1) Circle homeomorphisms - Poincare theory
- fixed and periodic points
- orienation preserving and reversing homeomorphisms
- rotation number of a homeomorphism
- existence of semi-conjugacy with a rotation
2) Circle diffeomorphisms - Denjoy theory
- existence of conjugacy
- Diophantine conditions
- theorem on regularity of the conjugacy
- counterexamples
Total student workload
30 h - lecture
4 h - exam
25 h - preparation for ongoing lectures, studying the literature
16 h - preparation for the exam
SUM: 75 h
Learning outcomes - knowledge
After finishing the course, student knows the elements of classical theory concerning circle diffeomorphisms, in particular Poincare and Denjoy theory.
(Code according to Polska Rama Kwalifikacji P8S_WG)
Learning outcomes - skills
After finishing the course, student is able to point out the conditions necessary for existence of the conugacy of a circle diffeomorphism with a rotation as well as deduce its regularity. Moreover, they are able to investigate basic dynamical and topological properties of circle homeomorphisms.
(Code according to Polska Rama Kwalifikacji P8S_UW and P8S_UO)
Learning outcomes - social competencies
After finishing the course student enhances their ability to solve complex problems and finding mathematical relations in everyfay life.
(Code according to Polska Rama Kwalifikacji P8S_KR)
Teaching methods
Traditional lecture. All notions and facts are illustrated by proper examples.
Expository teaching methods
- informative (conventional) lecture
Exploratory teaching methods
- classic problem-solving
Type of course
elective course
Prerequisites
Calculus I and II.
Course coordinators
Assessment criteria
Oral exam. Student needs to master the facts presented during the lecture.
Bibliography
"One-Dimensional Dynamics" - W. de Melo, S. van Strien, Springer-Verlag
"Groups of Circle Diffeomorphisms" - A. Navas, The University of Chicago Press
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: