Reversible Christoffel factorizations

TL;DR

This paper introduces reversible Christoffel factorizations, a new way to decompose Sturmian words into Christoffel words, revealing their structure through Abelian equivalence and three-interval exchange transformations.

Contribution

It defines reversible Christoffel factorizations of Sturmian words and characterizes their structure using the three gap theorem and interval exchange transformations.

Findings

Only 2 or 3 distinct Christoffel words occur in each RC factorization.

RC factorizations are either Sturmian words or derived from three-interval exchange transformations.

The study links Abelian equivalence, Christoffel words, and interval exchanges in Sturmian words.

Abstract

We define a family of natural decompositions of Sturmian words in Christoffel words, called *reversible Christoffel* (RC) factorizations. They arise from the observation that two Sturmian words with the same language have (almost always) arbitrarily long Abelian equivalent prefixes. Using the three gap theorem, we prove that in each RC factorization, only 2 or 3 distinct Christoffel words may occur. We begin the study of such factorizations, considered as infinite words over 2 or 3 letters, and show that in the general case they are either Sturmian words, or obtained by a three-interval exchange transformation.