Rotated odometers

Abstract

AbstractWe 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.