Module also offered within study programmes:
General information:
Name:
Ergodic theory
Course of study:
2019/2020
Code:
ZSDA-3-0254-s
Faculty of:
Szkoła Doktorska AGH
Study level:
Third-cycle studies
Specialty:
-
Field of study:
Szkoła Doktorska AGH
Semester:
0
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
prof. dr hab. Oprocha Piotr (oprocha@agh.edu.pl)
Dyscypliny:
matematyka
Module summary

The lecture is introduction to ergodic theory. The aim of classes is to increase understanding of basic properties and definitions presented during lecture, by solving short problems.

Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Social competence: is able to
M_K001 Student is able to apply advanced methods of ergodic theory to solve specific problems. SDA3A_U01 Activity during classes
Skills: he can
M_U001 Student has sufficient knowledge to understand research results at the edge of ergodic theory and dynamical systems. SDA3A_U02
Knowledge: he knows and understands
M_W001 Student knows main measure-theoretic tools used in modern theory of dynamical systems. SDA3A_W02
M_W002 Student is aware of main research directions in modern ergodic theory, and has sufficient knowledge to perform his own research in this field. SDA3A_W01
Number of hours for each form of classes:
Sum (hours)
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Prace kontr. przejść.
Lektorat
60 30 30 0 0 0 0 0 0 0 0 0
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Prace kontr. przejść.
Lektorat
Social competence
M_K001 Student is able to apply advanced methods of ergodic theory to solve specific problems. + + - - - - - - - - -
Skills
M_U001 Student has sufficient knowledge to understand research results at the edge of ergodic theory and dynamical systems. + + - - - - - - - - -
Knowledge
M_W001 Student knows main measure-theoretic tools used in modern theory of dynamical systems. + + - - - - - - - - -
M_W002 Student is aware of main research directions in modern ergodic theory, and has sufficient knowledge to perform his own research in this field. + + - - - - - - - - -
Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 107 h
Module ECTS credits 4 ECTS
Udział w zajęciach dydaktycznych/praktyka 60 h
Preparation for classes 30 h
Realization of independently performed tasks 10 h
Examination or Final test 2 h
Contact hours 5 h
Module content
Lectures (30h):

1. Measure preserving transformations. Poincare recurrence theorem.
2. Invariant and ergodic measures. Notions of mixing.
3. Koopman operator and von Neumann ergodic theorem.
4. Birkhoff ergodic theorem.
5. Isomorphisms and spectral properties.
6. Invariant measures for continuous maps.
7. Entropy (metric and topological). Variational principle.

Auditorium classes (30h):

Students will apply new techniques, theorems and constructions learned during lectures to solve a given problems.

Additional information
Teaching methods and techniques:
  • Lectures: Classical lecture on blackboard.
  • Auditorium classes: Presentation and discussion on correctness of solutions. Comparison of different solving techniques used by various students.
Warunki i sposób zaliczenia poszczególnych form zajęć, w tym zasady zaliczeń poprawkowych, a także warunki dopuszczenia do egzaminu:

Students solve exercises given on lecture and present own solutions during classes. On that basis students are evaluated during classes. Positive mark from classes entitles to oral exam.

Participation rules in classes:
  • Lectures:
    – Attendance is mandatory: Yes
    – Participation rules in classes: Nie określono
  • Auditorium classes:
    – Attendance is mandatory: Yes
    – Participation rules in classes: Students present own solutions to problems provided during lecture.
Method of calculating the final grade:

Final grade is a weighted average of exam (0.5) classes (0.5)

Sposób i tryb wyrównywania zaległości powstałych wskutek nieobecności studenta na zajęciach:

The student should report to the lecturer to determine the individual way of catching up.

Prerequisites and additional requirements:

Basic knowledge of probability theory (probability measures) and mathematical analysis.

Recommended literature and teaching resources:

P. Walters, An introduction to ergodic theory, Springer-Verlag, New York-Berlin, 1982.

K. Petersen, Ergodic theory, Cambridge University Press, Cambridge, 1989.

W. Parry, Entropy and generators in ergodic theory,W. A. Benjamin, Inc., New York-Amsterdam, 1969.

M. Einsiedler, T. Ward, Ergodic Theory with a view towards Number Theory, Springer-Verlag 2010.

Scientific publications of module course instructors related to the topic of the module:
P. Oprocha, P. Potorski, P. Raith, “Mixing properties in expanding Lorenz maps”, Adv. Math., 343 (2019), 712-755.

J. Boroński, J. Kupka and P. Oprocha, “Mixing completely scrambled system exists”, Erg. Th. Dynam. Syst., 39 (2019), 62-73.

P. Oprocha, “Double minimality, entropy and disjointness with all minimal systems”, Discrete Contin. Dyn. Syst., 39 (2019), 263—275.

C. Good, P. Oprocha, M. Puljiz, “Shadowing, asymptotic shadowing and s-limit shadowing”, Fund. Math., 244 (2019), 287-312.

J. Boroński, A. Clark and P. Oprocha, “A compact minimal space Y such that its square YxY is not minimal”, Adv. Math., 335 (2018), 261-275.

W. Brian, P. Oprocha,“Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics”, Israel J. Math., 227 (2018), 423-453.

Y. Dong, P. Oprocha and X. Tian, “On the irregular points for systems with the shadowing property”, Erg. Th. Dynam. Syst.,28 (2018), 2108-2131.

Additional information:

None