## Geometric representation of sequences of complexity $$2n+1$$. (Représentation géométrique de suites de complexité $$2n+1$$.)(French)Zbl 0789.28011

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.

### MSC:

 28D05 Measure-preserving transformations 11K50 Metric theory of continued fractions
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### References:

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