Morphisms fixing words associated with exchange of three intervals

TL;DR

This paper investigates the morphisms that preserve 3iet words, which code exchange of three intervals, revealing connections with Sturmian words through the concept of amicability.

Contribution

It introduces the notion of amicability and characterizes morphisms that fix 3iet words, linking them to Sturmian word morphisms.

Findings

Identifies morphisms fixing 3iet words.

Establishes a connection between 3iet and Sturmian morphisms.

Provides a new framework using amicability.

Abstract

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.