Rotated Odometers

TL;DR

This paper introduces rotated odometers, a class of infinite interval exchange transformations, and explores their measure-theoretic, topological, and ergodic properties through Bratteli-Vershik systems and specific examples.

Contribution

It establishes a connection between rotated odometers and first return maps of rational flows on complex translation surfaces, providing new insights into their dynamics.

Findings

Rotated odometers are measurably isomorphic to first return maps of rational flows.

They exhibit diverse topological and ergodic properties.

Specific examples illustrate the range of behaviors of these transformations.

Abstract

We describe the infinite interval exchange transformations, called the rotated odometers, that 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.