Interval exchanges, admissibility and branching Rauzy induction

TL;DR

This paper introduces a new notion of admissibility for subintervals in interval exchange transformations, explores their codings, and characterizes transformations over quadratic fields, revealing their factor sets are primitive morphic.

Contribution

It defines admissibility for subintervals, proves properties of derived sets in interval exchange transformations, and characterizes transformations over quadratic fields as primitive morphic.

Findings

Derived sets of regular interval exchange sets are also regular with the same number of intervals.

Admissible intervals are characterized via a branching Rauzy induction.

Factor sets of transformations over quadratic fields are primitive morphic.

Abstract

We introduce a definition of admissibility for subintervals in interval exchange transformations. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular interval exchange set is a regular interval exchange set with the same number of intervals. Derivation is taken here with respect to return words. We characterize the admissible intervals using a branching version of the Rauzy induction. We also study the case of regular interval exchange transformations defined over a quadratic field and show that the set of factors of such a transformation is primitive morphic. The proof uses an extension of a result of Boshernitzan and Carroll.