A unified introduction to the dynamics of interval exchange maps and related topics, such as the geometry of translation surfaces, renormalization operators, and Teichmüller flows, starting from the basic definitions and culminating with the proof that almost every interval exchange map is uniquely ergodic. Great emphasis is put on examples and geometric interpretations of the main ideas.
We prove a new type of universality theorem for the Riemann zeta-function and other -functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.
The original version of the article was published in Central European Journal of Mathematics, 2005, 3(4), 591–605. Unfortunately, the original version of this article contains a mistake. We give some corrections to our work.
Due to a technical error, part of a sentence was omitted on the top of page 8. The first line should read: “where , or , means the number of folds of the covering ending at p, i.e. covering a neighbourhood of p in without covering p itself”.