Arnoux, Pierre; Rauzy, Gérard Geometric representation of sequences of complexity \(2n+1\). (Représentation géométrique de suites de complexité \(2n+1\).) (French) Zbl 0789.28011 Bull. Soc. Math. Fr. 119, No. 2, 199-215 (1991). Let \(A\) be a finite set and let \(\Omega= A^{\mathbb{N}}\) be all of the one sided infinite sequences over \(A\). For any \(u\in\Omega\) and \(n\in\mathbb{N}\) let \(L_ n(u)\) be the set of subwords of \(u\) of length \(n\) (\(v\in A^ n\) is a subword of \(u\) if for some \(k\) one has \(v_ 1= u_{k+1},\dots,v_ n= u_{k+n}\)), and let \(p_ n(u)\) be the cardinality of \(L_ n(u)\). The sequence \((p_ n(u))\) is called the complexity of \(u\). This paper studies minimal sequences \(u\) in \(\Omega\) of complexity \(n+1\) and \(2n+1\). The analysis uses the de Bruijn graph of \(L_ n(u)\) (where two words \(u\), \(v\) are connected if for some \(a,b\in A\) one has \(u= bw\) and \(v= wa\)). It is shown how all minimal sequences \(u\) of complexity \(p_ n(u)= n+1\) can be described by infinite sequences consisting of two substitutions, and how this leads to (the well-known) isomorphism of the action of the shift on the closed orbit of \(u\) with a rotation on the circle. For a class of the sequences of complexity \(p_ n(u)= 2n+1\) satisfying a regularity condition on their de Bruijn graphs it is shown that these generate closed orbits which are isomorphic to interval exchange transformations with six intervals. Reviewer: F.M.Dekking (Delft) Cited in 8 ReviewsCited in 180 Documents MSC: 28D05 Measure-preserving transformations 11K50 Metric theory of continued fractions Keywords:minimal sequences; complexity; de Bruijn graph; closed orbits; interval exchange transformations × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] ARNOUX (P.) . Un exemple de semi-conjugaison entre un échange d’intervalles et une rotation sur le tore , Bull. Soc. Math. France, t. 116, 1986 , p. 489-500. Numdam | MR 91a:58138 | Zbl 0703.58045 · Zbl 0703.58045 [2] ARNOUX (P.) et YOCCOZ (J.C.) . Construction de difféomorphismes pseudo-Anosov , C.R.A.S., t. 292, 1981 , p. 75-78. MR 82b:57018 | Zbl 0478.58023 · Zbl 0478.58023 [3] ITO (S.) et KIMURA (M.) . On Rauzy fractal , prépublication. · Zbl 0734.28010 [4] KEANE (M.) . Interval exchange transformations , Math. Z., t. 141, 1975 , p. 25-31. Article | MR 50 #10207 | Zbl 0278.28010 · Zbl 0278.28010 · doi:10.1007/BF01236981 [5] KRIEGER (W.) . On entropy and generators of measure-preserving transformations , Trans. Amer. Math. Soc., t. 149, 1970 , p. 453-464. MR 41 #3710 | Zbl 0204.07904 · Zbl 0204.07904 · doi:10.2307/1995407 [6] RAUZY (G.) . Suites à termes dans un alphabet fini , Séminaire de théorie des nombres de Bordeaux, t. 25, 1983 , p. 1-16. Article | MR 85j:11088 | Zbl 0547.10048 · Zbl 0547.10048 [7] RAUZY (G.) . Nombres algébriques et substitutions , Bull. Soc. Math. France, t. 110, 1982 , p. 147-178. Numdam | MR 84h:10074 | Zbl 0522.10032 · Zbl 0522.10032 [8] RAUZY (G.) . Rotations sur les groupes, nombres algébriques et substitutions , prépublication. · Zbl 0726.11019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.