Matrices of 3iet preserving morphisms

Abstract

We study matrices of morphisms preserving the family of words coding 3-interval exchange transformations. It is well known that matrices of morphisms preserving sturmian words (i.e. words coding 2-interval exchange transformations with the maximal possible factor complexity) form the monoid $\{\mathbf{M}\in {\mathbb{N}}^{2\times 2}|\det \mathbf{M}=\pm 1\}=\{\mathbf{M}\in {\mathbb{N}}^{2\times 2}|\mathbf{M}\mathbf{E}{\mathbf{M}}^T=\pm \mathbf{E}\}$, where $\mathbf{E}=(\begin{array}{ccc}0& 1-1& 0\end{array})$. We prove that in case of exchange of three intervals, the matrices preserving words coding these transformations and having the maximal possible subword complexity belong to the monoid $\{\mathbf{M}\in {\mathbb{N}}^{3\times 3}|\mathbf{M}\mathbf{E}{\mathbf{M}}^T=\pm \mathbf{E},\det \mathbf{M}=\pm 1\}$, where $\mathbf{E}=\left(\begin{array}{ccccccc}0& 1& 1-1& 0& 1-1& -1& 0\end{array}\right)$.

26 pages, 4 figures

Cited by in corpus (2)

• Characterization of substitution invariant 3iet words
• A note on 3iet preserving morphisms