# Dynamical systems (ergodic theory) (TUD)

Vak
2010-2011

Introduction to ergodic theory and its applications to number theory. To get an idea what the course is about, consider the following sequence 3/2, 9/4, 27/8, 81/16, 243/32, … and reduce that sequence modulo 1 to get 1/2, 1/4, 3/8, 1/16, 19/32, … Now prove that the reduced sequence is dense in the unit interval. If you find a proof, please tell no one but the lecturer. This sequence has to do with Littlewood’s conjecture. If you cannot find a proof right away, then you may consider another sequence 1, 2, 4, 8, 16, 32, 64, 128 and reduce that sequence by just taking the initial digit 1, 2, 4, 8, 1, 3, 6, 1, … Now try to find the frequence of the number 7 in that sequence. Is it higher than the frequency of the number 8? What you are looking for here, more or less, is an invariant measure on the circle R/Z that for a rotation that arises out of x -> 2x mod 1 and a jump. Both sequences involve ergodic transformations. The second transformation is uniquely ergodic. The first transformation is not and that is why the first sequence is harder to get a hold on. In this course we will study various types of ergodic transformations and aim to understand Furstenberg’s plan to attack the Littlewood conjecture.

Literature
Einsiedler and Ward – Ergodic Theory with an emphasis on Number Theory. Springer GTM, to appear in 2010. Contact the lecturer for details.
Vakcode TUD
WI4042
Prerequisites
Measure theory and basis functional analysis (Linear Analysis in Leiden)
Examination
Weekly exercises. Final oral exam or a written exam in the highly unlikely event that the number of candidates exceeds 10.
Contact
2 hours per week. You may have to do some lecturing yourself.
Remark
This course can be part of a Leiden master programme