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Zbl 0789.28011
Arnoux, Pierre; Rauzy, Gérard
Geometric representation of sequences of complexity $2n+1$. (Représentation géométrique de suites de complexité $2n+1$.)
(French)
[J] Bull. Soc. Math. Fr. 119, No.2, 199-215 (1991). ISSN 0037-9484

Let $A$ be a finite set and let $\Omega= A\sp{\bbfN}$ be all of the one sided infinite sequences over $A$. For any $u\in\Omega$ and $n\in\bbfN$ let $L\sb n(u)$ be the set of subwords of $u$ of length $n$ ($v\in A\sp n$ is a subword of $u$ if for some $k$ one has $v\sb 1= u\sb{k+1},\dots,v\sb n= u\sb{k+n}$), and let $p\sb n(u)$ be the cardinality of $L\sb n(u)$. The sequence $(p\sb n(u))$ 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 $L\sb n(u)$ (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 $p\sb n(u)= n+1$ 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 $p\sb n(u)= 2n+1$ 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.
[F.M.Dekking (Delft)]
MSC 2000:
*28D05 Measure-preserving transformations
11K50 Metric theory of continued fractions

Keywords: minimal sequences; complexity; de Bruijn graph; closed orbits; interval exchange transformations

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