Abstract
We introduce a multidimensional continued fraction algorithm based on Arnoux-Rauzy and Poincaré algorithms, and we study its associated S-adic system. An S-adic system is made of infinite words generated by the composition of infinite sequences of substitutions with values in a given finite set of substitutions, together with some restrictions concerning the allowed sequences of substitutions, expressed in terms of a regular language. We prove that these words have a factor complexity p(n) with lim sup p(n)/n < 3, which provides a proof for the convergence of the associated algorithm by unique ergodicity.
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References
Arnoux, P., Rauzy, G.: Représentation géométrique de suites de complexité 2n+1. Bull. Soc. Math. France 119(2), 199–215 (1991)
Berthé, V., Delecroix, V.: Beyond substitutive dynamical systems: s-adic expansions (preprint, 2013)
Berthé, V., Labbé, S.: Uniformly balanced words with linear complexity and prescribed letter frequencies. In: Proc. 8th Int. Conf. on Words. EPTCS, vol. 63, pp. 44–52. Open Publishing Association (2011)
Boshernitzan, M.: A unique ergodicity of minimal symbolic flows with linear block growth. J. Analyse Math. 44, 77–96 (1984/1985)
Cassaigne, J.: Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4(1), 67–88 (1997); Journées Montoises (Mons, 1994)
Cassaigne, J., Nicolas, F.: Factor complexity. In: Combinatorics, Automata and Number Theory. Encyclopedia Math. Appl., vol. 135, pp. 163–247. Cambridge Univ. Press, Cambridge (2010)
Dasaratha, K., Flapan, L., Garrity, T., Lee, C., Mihaila, C., Neumann-Chun, N., Peluse, S., Stroffregen, M.: Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms. arXiv:1208.4244 (2012)
Durand, F., Leroy, J., Richomme, G.: Do the Properties of an S-adic Representation Determine Factor Complexity? J. of Int. Seq. 16 (2013)
Ferenczi, S., Monteil, T.: Infinite words with uniform frequencies, and invariant measures. In: Combinatorics, Automata and Number Theory. Encycl. Math. Appl., vol. 135, pp. 373–409. Cambridge Univ. Press (2010)
Garrity, T.: On periodic sequences for algebraic numbers. J. Number Theory 88(1), 86–103 (2001)
Klouda, K.: Bispecial factors in circular non-pushy D0L languages. Theoret. Comput. Sci. 445, 63–74 (2012)
Labbé, S.: Structure des pavages, droites discèrtes 3D et combinatoire des mots. PhD thesis, Université du Québec à Montréal (May 2012)
Leroy, J.: Some improvements of the S-adic conjecture. Adv. in Appl. Math. 48(1), 79–98 (2012)
Nogueira, A.: The three-dimensional Poincaré continued fraction algorithm. Israel J. Math. 90(1-3), 373–401 (1995)
Schweiger, F.: Multidimensional continued fractions. Oxford Science Publications. Oxford University Press, Oxford (2000)
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Berthé, V., Labbé, S. (2013). Convergence and Factor Complexity for the Arnoux-Rauzy-Poincaré Algorithm. In: Karhumäki, J., Lepistö, A., Zamboni, L. (eds) Combinatorics on Words. Lecture Notes in Computer Science, vol 8079. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40579-2_10
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DOI: https://doi.org/10.1007/978-3-642-40579-2_10
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