On a faithful representation of Sturmian morphisms

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 define a faithful representation by (3×3)-matrices with integer entries. We find three convex cones in R3 and show that a matrix R∈Sl(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.