Codages de rotations et phénomènes d'autosimilarité
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 351-386.

Nous étudions une classe de suites symboliques, les codages de rotations, intervenant dans des problèmes de répartition des suites (nα)n∈ℕ et représentant une généralisation géométrique des suites sturmiennes. Nous montrons que ces suites peuvent être obtenues par itération de quatre substitutions définies sur un alphabet à trois lettres, puis en appliquant un morphisme de projection. L’ordre d’itération de ces applications est gouverné par un développement bi-dimensionnel de type “fraction continue” vérifiant un théorème de Lagrange. Nous utilisons ensuite cette propriété pour caractériser les codages de rotations faisant intervenir des phénomènes d’autosimilarité, puis en déduire une propriété de déséquilibre du langage de ces codages.

The paper focus on a class of symbolic sequences obtained by encoding rotations and offering a geometric framework for the study of generalizations of Sturmian sequences. Those symbolic sequences also appear in problems related to the uniform distribution of the sequences (nα)n∈ℕ. We show that they can be computed by iterating four different substitutions over a three-letter alphabet, followed by an appropriate projection. The iteration schema is governed by a two-dimensional continued fraction algorithm satisfying a full Lagrange type theorem. This property is used to characterize the subset of sequences having a self-similar structure and then to deduce a quantitative unbalance property for these particular codings.

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     title = {Codages de rotations et ph\'enom\`enes d'autosimilarit\'e},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux I},
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Adamczewski, Boris. Codages de rotations et phénomènes d'autosimilarité. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 351-386. https://www.numdam.org/item/JTNB_2002__14_2_351_0/

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