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Matrices of 3-iet preserving morphisms. (English) Zbl 1161.68042

Summary: 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 \(\{\mathbf M \in \mathbb N^{2 \times 2} \mid \text{det } \mathbf M = \pm 1 \} = \{ \mathbf M \in \mathbb N^{2 \times 2} \mid \mathbf {MEM} ^T = \pm \mathbf E \}\), where \(\mathbf E = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}\).
We prove that in the case of exchange of three intervals, the matrices preserving words coding these transformations and having the maximal possible subword complexity belong to the monoid \(\{ \mathbf M \in \mathbb N^{3 \times 3} \mid \mathbf {MEM}^T = \pm \mathbf E, \text{ det } \mathbf M = \pm 1\}\) where \(\mathbf E = \begin{pmatrix} 0&1&1\\ -1&0&1\\ -1&-1&0 \end{pmatrix}\).

MSC:

68R15 Combinatorics on words
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