2015
DOI: 10.1016/j.aam.2014.11.001
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Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincaré algorithm

Abstract: Mathematics Subject Classifications: 68R15, 37B10. AbstractThe Arnoux-Rauzy-Poincaré multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincaré algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in subtracting the sum of the two smallest entries to the largest if possible (Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the median and the median to the largest (the Poincaré step), and by performing… Show more

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Cited by 17 publications
(11 citation statements)
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“…In recent years, multidimensional continued fraction algorithms were used to obtain ternary balanced sequences with low factor complexity for any given letter frequency vector. Indeed the Brun algorithm leads to balanced sequences [10] and it was shown that the Arnoux-Rauzy-Poincaré algorithm leads to sequences of factor complexity p(n) ≤ 5 2 n + 1 [5]. In 2015, the first author introduced a new Multidimensional Continued Fraction algorithm [9] based on the study of Rauzy graphs.…”
Section: Introductionmentioning
confidence: 99%
“…In recent years, multidimensional continued fraction algorithms were used to obtain ternary balanced sequences with low factor complexity for any given letter frequency vector. Indeed the Brun algorithm leads to balanced sequences [10] and it was shown that the Arnoux-Rauzy-Poincaré algorithm leads to sequences of factor complexity p(n) ≤ 5 2 n + 1 [5]. In 2015, the first author introduced a new Multidimensional Continued Fraction algorithm [9] based on the study of Rauzy graphs.…”
Section: Introductionmentioning
confidence: 99%
“…Note that S bLSP ({a, b, c}) = {L a , L b , L c , λ abc , λ acb , λ bac , λ bca , λ cab , λ cba }. Observe that these morphisms are the mirror morphisms of Arnoux-Rauzy and Poincaré morphisms (here f is a mirror morphism of g if f (a) is the mirror image or reversal of g(a) for all letters a) used by V. Berthé and S. Labbé [5].…”
Section: Some Basic Morphismsmentioning
confidence: 99%
“…Note that Poincaré does have an invariant measure equivalent to Lebesgue but it is not conservative. Finally, the domain for AR-Poincaré [BL15] and for the additive version of Jacobi-Perron algorithm are fractals and defined by an IFS. Both of these algorithms are known to have an invariant measure absolutely continuous with respect to Lebesgue measure; this was shown for Jacobi-Perron algorithm by Broise and Guivarc'h [BAG01], and we will prove it for AR-Poincaré in a forthcoming paper, but we do not know their density function.…”
Section: Experimentations: Domain Of the Natural Extensionmentioning
confidence: 99%