Rotated odometers and actions on rooted trees
Volume 260 / 2023
Fundamenta Mathematicae 260 (2023), 233-249
MSC: Primary 37A05; Secondary 37E05, 28D05, 37B05, 37E25.
DOI: 10.4064/fm74-10-2022
Published online: 9 January 2023
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.
