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 | from | until | |
---|---|---|---|---|---|

Monday | 9:45--11:15 | SR11 | Lecture | 06.03.2023 | 26.06.2023 |

Thursday | 8:00--9:30 | SR11 | Lecture | 02.06.2023 | 29.06.2023 |

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)

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

- discrete time dynamical systems, including circle maps, doubling maps, continued fraction maps.
- Invariant measures, Krylov-Bogul'jubov Theorem.
- Absolute continuity, densities (= Radon-Nikodym derivative)
- Ergodicity and unique ergodicity
- Ergodic Theorems (Birkhoff, von Neumann) and basic applications.
- Normal numbers and Benford's Theorem
- Poincare's Recurrence Theorem, Kac's Lemma.
- Mixing and transfer operators
- shift spaces and Bernoulli meaures
- Perron-Frobenius Theorem
- Cellular automata

- Jane Hawkins, Ergodic Dynamics; From Basic Theory to Applications, Springer-Verlag2021, ISBN: 978-3-030-59242-4 (ebook) or ISBN: 978-3-030-59244-8 (paper back)
- Peter Walters, An Introduction to Ergodic Theory, Springer-Verlag 1975 ISBN 0-387-95152-0.
- Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete 8. Springer-Verlag, Berlin, 1987. ISBN: 3-540-15278-4
- Daniel Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, Clarendon Press Oxford 1990 ISBN 0-19-853572-4
- Karl Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 1983, Cambridge University Press ISBN 0-521-38997-6
- Michael Brin and Garrett Stuck, Introduction to Dynamical Systems, Cambridge University Press 2002, ISBN 0-521-80841-3
- Omri Sarig, Lecture Notes on Ergodic Theory Penn State, Fall 2008, in .pdf
- Mike Hochman
- Charles Walkden

Week | Lecturer | Topic | Remarks |
---|---|---|---|

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 Theorem | Hawkins 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 Perron-Frobenius | 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 |

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

Material:

- discrete time dynamical systems, including circle maps, doubling maps, continued fraction maps.
- Invariant measures, Krylov-Bogul'jubov Theorem.
- Absolute continuity, densities (= Radon-Nikodym derivative)
- Ergodicity and unique ergodicity
- Ergodic Theorems (Birkhoff, von Neumann) and basic applications.
- Normal numbers and Benford's Theorem
- Poincare's Recurrence Theorem, Kac's Lemma.
- Mixing and transfer operators
- shift spaces and Bernoulli meaures
- Perron-Frobenius Theorem
- Cellular automata

- Class notes in pdf. The material up to and including Section 12 in these notes roughly covers the material that we covered (although the main text was the book by Hawkins). NB: These notes don't cover: the Perron-Frobenius Theorem and countable spectrum of the Koopman operator, multiple mixing, Rohlin partitions, equidistributions and Weyl's criterion... This set of notes may still be updated, and corrected.
- Julia sets as pinched disk models: The Basilica (angle 1/3) and Douady's Rabbit (angle 1/7)

Updated June 22 2023