Definition 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.
Objective
The 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.
No 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)
The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examination
oral 20 minutes
Additional information on mode of examination
20 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.
Performance assessment as a semester course