Rotated Odometers and Actions on Rooted Trees

TL;DR

This paper studies rotated odometers, a type of infinite interval exchange transformation, showing their connection to actions on rooted trees and the adding machine, with implications for group actions on binary trees.

Contribution

It establishes a measurable isomorphism between rotated odometers with dyadic intervals and actions on rooted trees, highlighting the role of the adding machine.

Findings

Rotated odometers are measurably isomorphic to $bZ$-actions on rooted trees.

The minimal aperiodic subsystem corresponds to the adding machine.

Applications to group actions on binary trees are discussed.

Abstract

A rotated odometer is an infinite interval exchange transformation (IET) obtained as a composition of the von Neumann-Kakutani map and a finite IET of intervals of equal length. In this paper, we consider rotated odometers for which the finite IET is of intervals of length $2^{-N}$, for some $N \geq 1$. We show that every such system is measurably isomorphic to a $\mathbb{Z}$-action on a rooted tree, and that the unique minimal aperiodic subsystem of this action is always measurably isomorphic to the action of the adding machine. We discuss the applications of this work to the study of group actions on binary trees.