On the rigidity of Arnoux-Rauzy words

TL;DR

This paper proves that all Arnoux-Rauzy words are rigid, meaning they are fixed by powers of a single substitution, by analyzing their normal forms and shared primitive substitutions.

Contribution

It establishes the rigidity of all Arnoux-Rauzy words, extending known results from Sturmian and characteristic Arnoux-Rauzy words, using normalization of episturmian substitutions.

Findings

All Arnoux-Rauzy words are rigid.

Primitive substitutions fixing an Arnoux-Rauzy word share a common power.

Normalization of episturmian substitutions is key to the proof.

Abstract

An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of a same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words are rigid. The proof relies on two main ingredients: firstly, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.