MU03049 Dynamical Systems I

Mathematical Institute in Opava
Winter 2023
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Jana Hantáková, Ph.D. (lecturer)
doc. RNDr. Jana Hantáková, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Jana Hantáková, Ph.D.
Mathematical Institute in Opava
Timetable
Wed 8:05–9:40 117
  • Timetable of Seminar Groups:
MU03049/01: Fri 8:05–9:40 117, J. Hantáková
Prerequisites (in Czech)
TYP_STUDIA(BN)
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
This course is a graduate level introduction to the mathematical theory of discrete dynamical systems on compact metric spaces and additionaly on the intercal. We will discuss some fundamental examples in the field, including circle rotations, shifts and subshifts, quadratic maps. We cover topics such as limit sets and recurrence, topological mixing, transitivity, entropy and symbolic dynamics.
Syllabus
  • 1. Basic definition
    Orbit (full, forward and backward). Periodic orbit.
    Šarkovsky theorem.
    Critical point, hyperbolic, attractive, repulsive.
    2. Examples of dynamic systems
    Quadratic system - logistic function, tent map, irrational rotation.
    3. Symbolic dynamics
    Shift map and its basic properties. Finite type shift.
    4. Topological dynamics
    Minimal set, omega-limit set, non-wandering set, conjugation.
    Properties of dynamic systems - transitivity, mixing, sensitivity.
    Recurrent and uniformly recurrent point.
    Topological entropy, chaos.
Literature
    required literature
  • L. S. Block, W. A. Coppel. Dynamics in one dimension. Lecture Notes in Mathematics, 1513. Springer-Ver, 1992. info
  • R. L. Devaney. An introduction to chaotic dynamical systems. Second edition, 1989. info
  • P. Walters. An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verl, 1982. info
    recommended literature
  • J. Smítal. On functions and functional equations. Adam Hilger, Ltd., Bristol, 1988. ISBN 0-85274-418-8. info
  • H.Furstenberg. Recurrence in Ergodic Theory and Combinational Number Theory. Princeton University Press, Princeton, New Jersy, 1981. info
Assessment methods
Course credit will be awarded for active participation in the exercise, the student will have to demonstrate an understanding of the theory and basic concepts using specific examples. Attendance in exercises is mandatory. The final exam will be oral. The student chooses one of the discussed topics at random and, after preparation, will have to show knowledge of basic definitions, statements, and their proofs.
Language of instruction
Czech
Further Comments
Study Materials
The course can also be completed outside the examination period.
The course is also listed under the following terms Winter 2014, Winter 2015, Winter 2016, Winter 2017, Winter 2018, Winter 2019, Winter 2020, Winter 2021, Winter 2022, Winter 2024.
  • Enrolment Statistics (Winter 2023, recent)
  • Permalink: https://is.slu.cz/course/sumu/winter2023/MU03049