Matrices of 3iet preserving morphisms

2007/02/12 by Petr Ambrož, P. Ambroz, Z. Masáková +6
Computer Science · Mathematics · #15A36 #68R15 #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.CO #msc:15A36 #msc:68R15 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0702336

26 pages, 4 figures

arxiv created 2007/02/12 · openalex publication_date 2007/02/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \\boldsymbolM∈ℕ^{2× 2} | det\boldsymbolM=±1\ = \\boldsymbolM∈ℕ^{2× 2} | \boldsymbolM\boldsymbolE\boldsymbolM^T = ±\boldsymbolE\, where \boldsymbolE = (\beginsmallmatrix0&1 -1&0\endsmallmatrix). 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 \\boldsymbolM∈ℕ^{3× 3} | \boldsymbolM\boldsymbolE\boldsymbolM^T = ±\boldsymbolE, det\boldsymbolM=± 1\, where \boldsymbolE = (\beginsmallmatrix0&1&1 -1&0&1 -1&-1&0\endsmallmatrix).

Citations

• One-dimensional Substitution Tilings with an Interval Projection Structure
• APPROXIMATIONS IN ERGODIC THEORY
• Symbolic Dynamics II. Sturmian Trajectories

Related

• A note on 3iet preserving morphisms
• Morphisms fixing words associated with exchange of three intervals
• Exchange of three intervals: itineraries, substitutions and\n palindromicity
• Characterization of substitution invariant 3iet words
• On the number of factors in codings of three interval exchange