Matrices of 3-iet preserving morphisms

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 {M ∈ N | det M = ±1} = {M ∈ N | MEM = ±E}, where E = ( 0 1 −1 0 ). 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 {M ∈ N | MEM = ±E, det M = ±1}, where E = ( 0 1 1 −1 0 1 −1 −1 0 ) .