Rotated odometers

Abstract

We describe the infinite interval exchange transformations, called the rotated odometers, which are obtained as compositions of finite interval exchange transformations and the von Neumann-Kakutani map. We show that with respect to Lebesgue measure on the unit interval, every such transformation is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends. We describe the dynamics of rotated odometers by means of Bratteli-Vershik systems, derive several of their topological and ergodic properties, and investigate in detail a range of specific examples of rotated odometers.