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Zbl 1113.37003
Adamczewski, Boris
Rotation encoding and self-similarity phenomenon. (Codages de rotations et phénomènes d'autosimilarité.) (French)
[J] J. Théor. Nombres Bordx. 14, No. 2, 351-386 (2002). ISSN 1246-7405

The author introduces codings of rotations as a tool to study problems related to the uniform distribution of sequences $(n\alpha)$. Codings of rotations are obtained by coding the orbit on the circle $\Pi_1$ cut into two intervals $[0,\beta[ \cup [\beta,1[$, of a point $x$ under the action of an irrational rotation of angle $\alpha$. A special case is well understood, that is, when the length of one interval equals the angle of rotation. Then, sequences are called Sturmian; among their remarkable properties, their language is completly determined by the continued fraction expansion of the angle of rotation $\alpha$. In this paper, the author focuses on non degenerated sequences ($\beta \not\in {\Bbb Z}+\alpha{\Bbb Z}$ and $\alpha \not \in {\Bbb Q}$), coding the so-called i.d.o.c exchanges of three intervals. By using an induction process, he proves that such sequences are obtained as a two-letters projection of the iteration of four specific three-letters substitutions. The order of the substitutions defines a two-dimensional continued fraction expansion of the parameters $(\alpha,\beta)$ of the rotation. The author proves that this expansion is ultimately periodic if and only if the parameters belong to a same quadratic field (Lagrange type theorem). In this case, the equilibrium function of the sequence is not bounded, implying that the language of some non-degenerated codings of rotation is irregular. This shows an intrinsic difference of behavior with the degenerated case $\beta \in {\Bbb Z}+\alpha{\Bbb Z}$.
[Anne Siegel (Rennes)]

MSC 2000:: *37B10 Symbolic dynamics
37A45 Relations of ergodic theory with number theory and harmonic analysis
37E10 Maps of the circle
11B85 Automata sequences

Keywords: coding of rotation; Sturmian sequences; uniform distribution; substitution; continued fraction

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