VO 250138-1 Ergodic Theory I (6 ECTS)

Lecturer: Prof. Henk Bruin and Dr. Davide Ravotti

Email Henk Bruin or Davide Ravotti for further information for this course.


Thursday June 22 was the last day of the course. For the oral exam, please contact Davide Ravotti for appointments untl July 7 and Henk Bruin for after July 10.


Day Time Room fromuntil
Monday 9:45--11:15 SR11 Lecture06.03.202326.06.2023
Thursday 8:00--9:30 SR11 Lecture02.06.202329.06.2023

No classes on: March 13 (Rektorstag), April 3-14 (Easter break), May 1st (Labour day), May 18 (Ascension day), May 29 (Pentacost), June 8 (Corpus Christi)

Contents of the course

This is an introduction to ergodic theory, that is: the study of how invariant measures play a role in dynamical systems. Topics to be discussed are likely to include

The course will be given in English


Week LecturerTopicRemarks
Week 1
Thursday March 2 Bruin Introduction
Week 2
Monday March 6 Bruin Poincaré Recurrence
Kac Lemma
Hawkins Chapter 2
Thursday March 9 Bruin Koopman operator
and its spectrum
Bernoulli shifts
Hawkins Chapter 4.1
Week 3
Monday March 13 Rectorsday
Thursday March 16 Ravotti Von Neuman Ergodic Theorem Hawkins Chapter 4.2
Week 4
Monday March 20 Ravotti Maximal Ergodic Theorem Hawkins Chapter 4.3
Thursday March 23 Bruin Birkhoff Ergodic TheoremHawkins Chapter 4.3
Week 5
Monday March 27 Ravotti Spectrum of the Koopman operator
Unique ergodicity: definition and basic properties
Hawkins Chapter 4.4-5
Thursday March 30 Ravotti characterizations of unique ergodicity.
Irrational rotations are uniquely ergodic.
Equidistribution mod 1 and Weyl's Criterion
Hawkins Chapter 4.5
Week 6
Monday April 17 Bruin Circle rotations (Koksma-Denjoy)
Normal Numbers
Benford's Law
Hawkins Chapter 4.6
Thursday April 20 Bruin Continued fractions
Gauss map
Week 7
Monday April 24 Ravotti weak mixing, mixing
multiple mixing
doubling map is mixing
Hawkins Chapter 5.1
Thursday April 27 Ravotti Characterizations of weak mixing Hawkins Chpater 5.1
Week 8
Monday May 1 Labour Day
Thursday May 4 Ravotti Rohlin partitions Hawkins Chapter 5.2
Week 9
Monday May 8 Bruin Exercises List of exercises
Thursday May 11 Bruin Exactness Hawkins Chapter 5.5
Week 9
Monday May 15 Bruin Transfer operator
Thursday May 18 Ascension Day
Week 9
Monday May 22 Bruin Bernoulli shifts
Koopman operator
and its spectrum
Hawkins Chapter 6.1
Thursday May 25 no class
Week 10
Monday May 29 Pentacost
Thursday June 1 Bruin Class cancelled
Week 11
Monday June 5 Bruin Markov shifts Hawkins Chapter 6.2
Thursday June 8 Corpus Christi
Week 12
Monday June 12 Bruin Perron-Frobenius Theorem Hawkins Chapter 7.1-2
Thursday June 15 Ravotti Applications
Hawkins Chapter 7.3
Week 13
Monday June 19 Ravotti Existence and bsolute continuity
of invariant measures
induced maps
Construction and ergodicity
Hawkins Chapter 8.1-2
Thursday June 22 Bruin Examples of induced maps
Chebyshev polynomial
Boole's Transformation
Hawins Chapter 8.3
Week 14
Monday June 26 Ravotti No class
Thursday June 29 Ravotti No class

  • List of exercises


    Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich).


    Course material (Hand-outs)

    Updated June 22 2023