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 ℚ-vector space spanned by the lengths of the exchanged subintervals. We prove that if T satisfies Keane’s infinite distinct orbit condition and rank(T) > 1 + [m/2], 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.