Interval exchanges, admissibility and branching Rauzy induction

Abstract

We introduce a definition of admissibility for subintervals in interval exchange transformations. We characterize the admissible intervals using a branching version of the Rauzy induction. 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 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.