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Cassaigne, Julien; Ferenczi, Sébastien; Zamboni, Luca Q.

Imbalances in Arnoux-Rauzy sequences. (English) Zbl 1004.37008

Ann. Inst. Fourier 50, No. 4, 1265-1276 (2000).
Arnoux-Rauzy sequences were introduced in [P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexité \(2n+1\), Bull. Soc. Math. Fr. 119, No. 2, 199-215 (1991; Zbl 0789.28011)]. The authors formulated at this time the conjecture that these sequences can be obtained as codings of rotations on the two-dimensional torus. This conjecture is disproved in the paper under review, where the authors actually disprove a conjecture of X. Droubay, J. Justin and G. Pirillo [Theor. Comput. Sci. 255, No. 1-2, 539-553 (2001; Zbl 0981.68126)] by showing that these sequences are not necessarily \(N\)-balanced: recall that a sequence is \(N\)-balanced for some integer \(N\) if for each letter the numbers of occurrences of this letter in any two words in the sequence having the same length differ by at most \(N\). Sturmian sequences are known to be 1-balanced.
Note that several papers in the bibliography have appeared:
[3] in Bull. Belg. Math. Soc., Simon Stevenin 8, 181-207 (2001; Zbl 1007.37001);
[5] (with a slightly different title) in Discrete Math. 223, 27-53 (2002; Zbl 0970.68124);
[6] (with a slightly different title) in Trans. Am. Math. Soc. 353, 5121-5144 (2001; Zbl 1142.37302);
[11] in J. Théor. Nombrés Bordx. 13, 371-394 (2001; Zbl 1038.37010);
[12] in Theor. Comput. Sci. 255, 539-553 (2001; Zbl 0981.68126);
[15] in Ann. Inst. Fourier 51, 861-901 (2001; Zbl 1029.11036);
[22] in Acta Arith. 95, 195-224 (2000; Zbl 0968.28005);
[29] in Acta Arith. 95, 167-184 (2000; Zbl 0953.11007);
[30] in Acta Arith. 96, 261-278 (2001; Zbl 0973.11030).
Also note that [20] has appeared: Cambridge, 2002.
Reviewer: J.-P.Allouche (Orsay)

Cited in 1 Review
Cited in 29 Documents

MSC:

37B10 Symbolic dynamics
68R15 Combinatorics on words
11B83 Special sequences and polynomials

Keywords:

infinite words; codings of rotations; return times; bounded reaminder sets; balanced sequences; Arnoux-Rauzy sequences; Sturmian sequences

Citations:

Zbl 0789.28011; Zbl 0981.68126; Zbl 1007.37001; Zbl 0970.68124; Zbl 1142.37302; Zbl 1038.37010; Zbl 1029.11036; Zbl 0968.28005; Zbl 0953.11007; Zbl 0973.11030
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Full Text: DOI Numdam EuDML

Online Encyclopedia of Integer Sequences:

Apply the tribonacci morphism 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1} n times to 1, and concatenate the resulting string.
0-limiting word of the morphism 0->10, 1->20, 2->0.
Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1. Fixed point of the morphism 0-> 0201020, 1->1020, 2->201020 starting from a(1)=0.

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