Definition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Connection to coding theory. Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points.
Lernziel
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. Finally we will use these methods to classify certain natural homogeneous measures and use this to prove equidistribution of periodic points on compact quotients of hyperbolic surfaces.
Skript
Entropy book under construction, available online under Link
Voraussetzungen / Besonderes
Measure Theory will be very important. No prior knowledge of dynamical systems will be assumed. Doctoral students are welcome to attend the course but cannot take it for credit.
Leistungskontrolle
Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Die Leistungskontrolle wird in jeder Session angeboten. Die Repetition ist ohne erneute Belegung der Lerneinheit möglich.
Prüfungsmodus
mündlich 20 Minuten
Zusatzinformation zum Prüfungsmodus
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 2019 and Winter 2020.
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.
Diese Angaben können noch zu Semesterbeginn aktualisiert werden; verbindlich sind die Angaben auf dem Prüfungsplan.