MU03049 Dynamical Systems I

Mathematical Institute in Opava
Winter 2017
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Michal Málek, Ph.D. (lecturer)
doc. RNDr. Michal Málek, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Michal Málek, Ph.D.
Mathematical Institute in Opava
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. Elementary notions - orbit (full, forward, backward), fixed point, eventually fixed point, phase portrait, Brower fixed
    theorem, Sharkovskii ordering.
    2. Hyperbolicity - critical point, hyperbolic point, attractive and repulsice point.
    3. Quadratic system - logistic map, the tent map, rotations of the circle.
    4. Symbolical dynamics - shift space and shift map.
    5. Topological dynamics I. - minimal sets, limit sets, nonwandering sets, conjugacy.
    6. Topological dynamics II. - transitivity, total transitivity, mixings, their relations and relations to the dense orbit.
    7. Topological dynamics III. - recurrence and relations to ninimality.
    8. Topological dynamics IV. - topological entropy.
Literature
    recommended 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
  • J. Smítal. On functions and functional equations. Adam Hilger, Ltd., Bristol, 1988. ISBN 0-85274-418-8. info
  • P. Walters. An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verl, 1982. info
  • H.Furstenberg. Recurrence in Ergodic Theory and Combinational Number Theory. Princeton University Press, Princeton, New Jersy, 1981. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
Teacher's information
Credid: credit test
Final exam: final test and oral exam
The course is also listed under the following terms Winter 2014, Winter 2015, Winter 2016, Winter 2018, Winter 2019, Winter 2020, Winter 2021, Winter 2022, Winter 2023, Winter 2024.
  • Enrolment Statistics (Winter 2017, recent)
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