This course is an introduction to ergodic theory motivated by applications such as Furstenberg's proof of Szemerédi's theorem on arithmetic progressions and Weyl's theorem on equidistribution of polynomials. In the second half of the course, I will develop entropy theory with the aim of proving Rudolph's theorem on x2, x3 invariant measures. No previous knowledge of ergodic theory is assumed.
Lecture notes are available here. (Last updated: 27 Nov. Contains all lectured material and two non-examinable chapters.) Please send comments to: pv270 at dpmms.
1. without hints, 1. with hints
2. without hints, 2. with hints
3. without hints, 3. with hints
Literature: The course will not follow any particular book. The book M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, Springer, 2011 is a useful source for the fist part of the course. The authors of that book together with E. Lindenstrauss work on a subsequent volume that develps entropy theory, which is available here. In the interest of saving time, the course will avoid many of the powerful technical tools used in that book. However, it is highly recommended for those whose interest in the subject goes beyond the course material. Another nice book on ergodic theory that covers most of the lectures is K. Petersen, Ergodic Theory, CUP, 1989.
Last three years' exam papers are available here, here and here. Note, however, that the course material varies from year to year, in particular the last two questions from 2015 are not relevant this year.