On a faithful representation of Sturmian morphisms

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