## Rotation encoding and self-similarity phenomenon. (Codages de rotations et phénomènes d’autosimilarité.)(French)Zbl 1113.37003

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 ($$\beta \not\in {\mathbb Z}+\alpha{\mathbb Z}$$ and $$\alpha \not \in {\mathbb Q}$$), 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 $$\beta \in {\mathbb Z}+\alpha{\mathbb Z}$$.

### MSC:

 37B10 Symbolic dynamics 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 37E10 Dynamical systems involving maps of the circle 11B85 Automata sequences
Full Text:

### References:

 [1] Adamczewski, B., Répartitions des suites (nα)n∈N et substitutions. Acta Arith., à paraître. · Zbl 1060.11043 [2] Arnoux, P., Rauzy, G., Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France119 (1991), 199-215. · Zbl 0789.28011 [3] Berthé, V., Tijdeman, R., Balance properties of multi-dimensional words. Theoret. Comput. Sci.273 (2002), 197-224. · Zbl 0997.68091 [4] Boshernitzan, M.D., Carroll, C.R., An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields. J. Anal. Math.72 (1997), 21-44. · Zbl 0931.28013 [5] Cassaigne, J., Ferenczi, S., Zamboni, L.Q., Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier (Grenoble) 50 (2000), 1265-1276. · Zbl 1004.37008 [6] Coven, E.M., Hedlund, G.A., Sequences with minimal block growth. Math. Systems Theory7 (1973), 138-153. · Zbl 0256.54028 [7] Crisp, D., Moran, W., Pollington, A., Shiue, P., Substitution invariant cutting sequences. J. Théor. Nombres Bordeaux5 (1993), 123-137. · Zbl 0786.11041 [8] Didier, G., Codages de rotations et fractions continues. J. Number Theory71 (1998), 275-306. · Zbl 0921.11015 [9] Didier, G., Combinatoire des codages de rotations. Acta Arith.85 (1998), 157-177. · Zbl 0910.11007 [10] Dumont, J.-M., Thomas, A., Systèmes de numération et fonctions fractales relatifs aux substitutions. Theoret. Comput. Sci.65 (1989), 153-169. · Zbl 0679.10010 [11] Durand, F., A characterization of substitutive sequences using return words. Discrete Math.179 (1998), 89-101. · Zbl 0895.68087 [12] Graham, R.L., Covering the positive integers by disjoint sets of the form {[nα + β]: n = 1, 2, ... }. J. Combinatorial Theory Ser. A 15 (1973), 354-358. · Zbl 0279.10042 [13] Hubert, P., Suites équilibrées. Theoret. Comput. Sci.242 (2000), 91-108. · Zbl 0944.68149 [14] Keane, M., Interval exchange transformations. Math. Z.141 (1975), 25-31. · Zbl 0278.28010 [15] Kesten, H., On a conjecture of Erdõs and Szüsz related to uniform distribution mod 1. Acta Arith.12 (1966/1967), 193-212. · Zbl 0144.28902 [16] Lopez, L.-M., Narbel, P., DOL-systems and surface automorphisms. Mathematical foundations of computer science, 1998 (Brno), . 1450, Springer, Berlin, 1998, pp. 522-532. · Zbl 0914.68113 [17] Lopez, L.-M., Narbel, P., Substitutions from Rauzy induction (extended abstract). Developments in language theory (Aachen, 1999), World Sci. Publishing, River Edge, NJ, 2000, pp. 200-209. · Zbl 1013.68150 [18] Lopez, L.-M., Narbel, P., Substitutions and interval exchange transformations of rotation class. Theoret. Comput. Sci.255 (2001), 323-344. · Zbl 0974.68160 [19] Lothaire, M., Algebraic combinatorics on words. Cambridge University Press, 2002. · Zbl 1001.68093 [20] Morse, M., Hedlund, G.A., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math.62 (1940), 1-42. [21] Rauzy, G., Sequences defined by iterated morphisms. Sequences (Naples/Positano, 1988), Springer, New York, 1990, pp. 275-286. · Zbl 0955.28501 [22] Rauzy, G., Échanges d’intervalles et transformations induites. Acta Arith.34 (1979), 315-328. · Zbl 0414.28018 [23] Rote, G., Sequences with subword complexity 2n. J. Number Theory46 (1994), 196-213. · Zbl 0804.11023 [24] Schweiger, F., Multidimensional continued fractions. Oxford University Press, 2000. · Zbl 0981.11029 [25] Sós, V.T., On strong irregularities of the distribution of {nα} sequences. Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 685-700. · Zbl 0519.10047 [26] Veech, W.A., Interval exchange transformations. J. Analyse Math.33 (1978), 222-272. · Zbl 0455.28006 [27] Veech, W.A., Gauss measures for transformations on the space of interval exchange maps. Ann. of Math.115 (1982), 201-242. · Zbl 0486.28014 [28] Zorich, A., Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46 (1996), 325-370. · Zbl 0853.28007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.