Most Interval Exchanges Have No Roots

TL;DR

This paper investigates the conditions under which minimal interval exchange transformations can be expressed as powers of other transformations, establishing rank bounds and characterizations for 3-interval cases.

Contribution

It proves a new rank-based criterion for non-decomposability of minimal IETs and classifies certain minimal IETs with a single orbit of discontinuities.

Findings

Transformations with rank > 1 + floor(m/2) are not powers of other IETs.

For 3-interval IETs, being a power is equivalent to satisfying Keane's condition.

Classified minimal IETs with all discontinuities in a single orbit.

Abstract

Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged intervals. We prove that if $T$ is minimal and the rank of $T$ is greater than $1+\lfloor m/2 \rfloor$, then $T$ cannot be written as a power of another interval exchange. We also demonstrate that this estimate on the rank cannot be improved. In the case that $T$ is a minimal 3-interval exchange transformation, we prove a stronger result: $T$ cannot be written as a power of another interval exchange if and only if $T$ satisfies Keane's infinite distinct orbit condition. In the course of proving this result, we give a classification (up to conjugacy) of those minimal IETs whose discontinuities all belong to a single orbit.