On a faithful representation of Sturmian morphisms

TL;DR

This paper introduces a faithful matrix representation of Sturmian morphisms using 3x3 integer matrices, providing new tools to analyze Sturmian sequences and their fixed points.

Contribution

It defines a faithful matrix representation for Sturmian morphisms via convex cones in R^3, offering a novel approach to study their properties.

Findings

Characterization of Sturmian morphisms via invariant convex cones

Alternative proofs of known fixed point results

A new result on the square root of Sturmian sequences

Abstract

The set of morphisms mapping any Sturmian sequence to a Sturmian sequence forms together with composition the so-called monoid of Sturm. For this monoid, we defne a faithful representation by $(3\times 3)$-matrices with integer entries. We find three convex cones in $\mathbb{R}^3$ and show that a matrix $R \in Sl(\mathbb{Z},3)$ is a matrix representing a Sturmian morphism if the three cones are invariant under multiplication by $R$ or $R^{-1}$. This property offers a new tool to study Sturmian sequences. We provide alternative proofs of four known results on Sturmian sequences fixed by a primitive morphism and a new result concerning the square root of a Sturmian sequence.