401-3378-19L  Entropy in Dynamics

SemesterSpring Semester 2021
LecturersM. Einsiedler
Periodicitynon-recurring course
Language of instructionEnglish


401-3378-19 GEntropy in Dynamics
continues to take place online via Zoom till the end of the semester
4 hrs
Wed14:15-16:00HG E 1.1 »
Fri10:15-12:00HG G 26.5 »
M. Einsiedler

Catalogue data

AbstractDefinition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points. Measure rigidity for commuting maps on the circle group.
ObjectiveThe course will lead to a firm understanding of measure theoretic dynamical entropy and its applications within dynamics. We will start with the basic properties of (conditional) entropy, relate it to the question of effective coding techniques, discuss and prove the Shannon-McMillan-Breiman theorem that is also known as the ergodic theorem for entropy. Moreover, we will discuss a topological counter part and relate this topological entropy to the measure theoretic entropy by the variational principle. We will use these methods to classify certain natural homogeneous measures, prove equidistribution of periodic points on compact quotients of hyperbolic surfaces, and establish a measure rigidity theorem for commuting maps on the circle group.
Lecture notesEntropy book under construction, available online under
Prerequisites / NoticeNo prior knowledge of dynamical systems will be assumed but measure theory will be assumed and very important. Doctoral students are welcome to attend the course for 2KP.

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits8 credits
ExaminersM. Einsiedler
Typesession examination
Language of examinationEnglish
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examinationoral 20 minutes
Additional information on mode of examination20 minutes preparation and 20 minutes exam (one candidate prepares during the 20 minutes oral exam of the previous candidate). The exam is only offered in the examination sessions Summer 2021 and Winter 2022.

The active participation in the exercise class via presentations as a voluntary learning task will be graded and can improve the total course unit grade by up to 0.25 grade points. Students can still achieve the maximum grade of 6 in the course unit even if they only sit the final examination.
This information can be updated until the beginning of the semester; information on the examination timetable is binding.

Learning materials

Main linkEntropy book
Only public learning materials are listed.


No information on groups available.


There are no additional restrictions for the registration.

Offered in

Mathematics BachelorSelection: AnalysisWInformation
Mathematics MasterSelection: AnalysisWInformation