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.


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