## General information

Course type | AMUPIE |

Module title | Ergodic Theory Of Numbers |

Language | English |

Module lecturer | prof. UAM dr hab. William Mance |

Lecturer's email | william.mance@amu.edu.pl |

Lecturer position | Professor |

Faculty | Faculty of Mathematics and Computer Science |

Semester | 2021/2022 (summer) |

Duration | 60 |

ECTS | 6 |

USOS code | 06-DETNLM0 |

## Timetable

## Module aim (aims)

The goal of this course is to introduce undergraduate students to ergodic theory and number theory through basic examples of numeration systems. Ergodic theory has recently become a more important tool in modern mathematics and this course aims to familiarize students with some basic tools that may be useful whether they continue in academia or go into industry. Normally, a substantial amount of analysis would be required to study this topic, but we will develop all the tools we need in the course. This course will touch open topics related to continued fractions, normal numbers, ergodic theory, entropy, Diophantine approximation, symbolic dynamics, entropy, and others (time permitting). Furthermore, the light introduction to some of these topics will prepare students if they plan to continue to a masters or PhD program.

## Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Some familiarity with analysis and linear algebra. Most important are mathematical maturity and a solid understanding of proof writing.

## Syllabus

Week 1: Introduction

Week 2: Continued Fractions

Week 3: Measure Theory

Week 4: Other numeration systems

Week 5: Other numeration systems (continued)

Week 6: Ergodicity

Week 7: Ergodicity (continued)

Week 8: Ergodicity (continued)

Week 9: Systems obtained from other systems

Week 10: Applications to GLS and beta expansions

Week 11: Diophantine Approximation

Week 12: More on continued fractions

Week 13: Entropy

Week 14: Entropy (continued)

Week 15: Uniform Distribution

## Reading list

Main Text:

Ergodic Theory of numbers (Dajani and Kraaikamp)

Supplemental Texts (however, much more advanced):

An Introduction to Ergodic Theory (Walters)

Ergodic Theory: With a View Towards Number Theory (Einsiedler and Ward)

Uniform Distribution of Sequences (Niederreiter and Kuipers)