Prerequisites
A background in measure theory obtained from any elementary bachelor course in measure theory is necessary. In addition some elementary knowledge of undergraduate functional analysis and point set topology as covered for example in the first four chapters of the book Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1 by Walter Rudin would provide sufficient background for this course.
Aim of the course
This course aims to give students a first introduction into the mathematical field of ergodic theory, to make them familiar with the most fundamental notions and results in the area and to give them skills to start an investigation into the ergodic properties of a discrete time dynamical system. In this course the following concepts will be represented:
(1) The notion of measure preservingness (stationarity) with several interpretations and exam- ples. The Poincar ́e Recurrence Theorem. The notion of ergodicity (which is a weak notion of independence) and its characterizations, the notion of conservativity (for infinite measure preserv ing systems).
(2) Ergodic Theorems (generalizations of the Strong Law of Large Numbers), such as Birkhoff’s and Von Neumann’s Ergodic Theorems. Some consequences of the Ergodic Theorems and the notions of weakly and strongly mixing.
(3) Isomorphism, factor maps and natural extensions.
(4) Some worked examples with a probabilistic and number theoretic flavor: Bernoulli and Markov shifts, continued fractions, normal numbers.
(5) The notion of entropy, the Shannon-Mcmillan-Breiman Theorem, and Lochs’ Theorem.
(6) Construction of invariant and ergodic measures for continuous transformations and unique ergodicity.
(7) The Perron-Frobenius operator and the existence of absolutely continuous invariant measures.
(8) Introduction to infinite ergodic theory, infinite ergodic theorems, induced and jump transformations
Rules about Homework/Exam
The grade of the course is determined by four sets of hand-in exercises (40%), presentations (20%) and a written exam (40%). The grade of the written exam must be at least 5 to possibly pass the course. Homework and presentation grades still count as part of the grade after the retake.
Lecture Notes/Literature
Book: A first course in ergodic theory by. K. Dajani and C. Kalle, Chapman and Hall/CRC, ISBN 9781000402773.
Name and institute lecturers: Karma Dajani (Utrecht University) and Charlene Kalle (Leiden University)
- Docent: Karma Dajani
- Docent: Charlene Kalle