Centralizers in the Group of Interval Exchange Transformations

TL;DR

This paper investigates the centralizer structure of interval exchange transformations, proving that under certain conditions, the only transformations commuting with a given one are its powers, with specific results for minimal 3-interval cases.

Contribution

It establishes conditions under which the centralizer of an interval exchange transformation is trivial, extending understanding of their algebraic structure.

Findings

Transformations with rank > 1 + floor(m/2) have trivial centralizers.

Minimal 3-interval exchange transformations have trivial centralizers if and only if they satisfy Keane's condition.

Provides a characterization of centralizers in the group of interval exchange transformations.

Abstract

We study the group of interval exchange transformations. 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 subintervals. We prove that if $T$ satisfies Keane's infinite distinct orbit condition and $\text{rank}(T)>1+\lfloor m/2 \rfloor$ then the only interval exchange transformations which commute with $T$ are its powers. In the case that $T$ is a minimal 3-interval exchange transformation, we prove a more precise result: $T$ has a trivial centralizer in the group of interval exchange transformations if and only if $T$ satisfies the infinite distinct orbit condition.