MU03049 Dynamical Systems I

Mathematical Institute in Opava
Winter 2020
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
doc. RNDr. Michal Málek, Ph.D. (lecturer)
doc. RNDr. Michaela Mlíchová, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Michal Málek, Ph.D.
Mathematical Institute in Opava
Tue 14:45–16:20 207
  • Timetable of Seminar Groups:
MU03049/01: Thu 9:45–11:20 118, M. Mlíchová
Prerequisites (in Czech)
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.
  • 1. Elementary notions
    Orbit (full, forward, backward), periodic orbit.
    Brower fixed point theorem.
    Sharkovskii ordering.
    2. Hyperbolicity
    Critical point, hyperbolic point, attractive and repulsice point.
    3. Examples of dynamical systems
    Quadratic system - logistic map, the tent map, rotations of the circle.
    4. Symbolical dynamics - shift space
    Shift map and its properties, shift of finite type.
    5. Topological dynamics
    Minimal sets, limit sets, nonwandering sets, centre, conjugacy.
    Transitivity, total transitivity, mixings, their relations and relations to the dense orbit.
    Recurrence and relations to ninimality.
    Topological entropy.
    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
Language of instruction
Further comments (probably available only in Czech)
Study Materials
The course can also be completed outside the examination period.
Teacher's information
Course credit: ability to verify notion on given examples
Final exam: knowledge of basic notions and assertions, at least partial understanding of theory
The course is also listed under the following terms Winter 2014, Winter 2015, Winter 2016, Winter 2017, Winter 2018, Winter 2019.
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