Languages of general interval exchange transformations

TL;DR

This paper explores the hierarchy of languages generated by various classes of interval exchange transformations, providing criteria to distinguish them and extending results to transformations with flips.

Contribution

It introduces a hierarchy of language classes for interval exchange transformations and characterizes each class with combinatorial criteria, including transformations with flips.

Findings

Languages form a strict hierarchy across transformation types.

Criteria for identifying each language class are established.

Results extend to interval exchanges with flips.

Abstract

The languages generated by interval exchange transformations have been characterized by Ferenczi-Zamboni (2008) and Belov-Cernyatev (2010) under some extra conditions on the system. Lifting these conditions leads us to consider successively natural codings of standard interval exchange transformations, natural codings of affine interval exchange transformations, grouped codings of affine interval exchange transformations, and natural codings of generalized interval exchange transformations. We show that these four classes of languages are strictly increasing, and give necessary and/or sufficient (but not all equally explicit) combinatorial criteria to describe each of them. These work also, mutatis mutandis, for interval exchanges with flips