Abstract
Given a finite set S of unimodular Pisot substitutions, we provide a method for characterizing the infinite sequences over S that allow to generate a full discrete plane when, starting from a finite seed, we iterate the multidimensional dual substitutions associated with S. We apply our results to study the substitutions associated with the Brun multidimensional continued fraction algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnoux, P., Ito, S.: Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8(2), 181–207 (2001)
Avila, A., Delecroix, V.: Pisot property for the Brun and fully subtractive algorithms (preprint, 2013)
Berthé, V., Jamet, D., Jolivet, T., Provençal, X.: Critical connectedness of thin arithmetical discrete planes. In: Gonzalez-Diaz, R., Jimenez, M.-J., Medrano, B. (eds.) DGCI 2013. LNCS, vol. 7749, pp. 107–118. Springer, Heidelberg (2013)
Berthé, V., Jolivet, T., Siegel, A.: Substitutive arnoux-rauzy sequences have pure discrete spectrum. Unif. Distrib. Theory 7(1), 173–197 (2012)
Berthé, V., Jolivet, T., Siegel, A.: Connectedness of Rauzy fractals associated with Arnoux-Rauzy substitutions (preprint, 2013)
Berthé, V., Siegel, A.: Tilings associated with beta-numeration and substitutions. Integers 5(3), A2, 46 (2005)
Berthé, V., Siegel, A., Thuswaldner, J.M.: Substitutions, Rauzy fractals, and tilings. In: Combinatorics, Automata and Number Theory, Encyclopedia of Mathematics and its Applications, vol. 135. Cambridge University Press (2010)
Berthé, V., Steiner, W., Thuswaldner, J.M.: Tilings with S-adic Rauzy fractals (preprint, 2013)
Brun, V.: Algorithmes euclidiens pour trois et quatre nombres. In: Treizième Congrès des Mathèmaticiens Scandinaves, Tenu à Helsinki, Août 18-23, pp. 45–64 (1957)
Fernique, T.: Multidimensional Sturmian sequences and generalized substitutions. Internat. J. Found. Comput. Sci. 17(3), 575–599 (2006)
Fernique, T.: Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 42(10), 2229–2238 (2009)
Fogg, N.P.: Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics, vol. 1794. Springer, Berlin (2002)
Frougny, C., Solomyak, B.: Finite beta-expansions. Ergodic Theory Dynam. Systems 12(4), 713–723 (1992)
Furukado, M., Ito, S., Yasutomi, S.I.: The condition for the generation of the stepped surfaces in terms of the modified Jacobi-Perron algorithm (preprint, 2013)
Ito, S., Ohtsuki, M.: Parallelogram tilings and Jacobi-Perron algorithm. Tokyo J. Math. 17(1), 33–58 (1994)
Ito, S., Rao, H.: Atomic surfaces, tilings and coincidence. I. Irreducible case. Israel J. Math. 153, 129–155 (2006)
Paysant-Le Roux, R., Dubois, E.: Une application des nombres de Pisot à l’algorithme de Jacobi-Perron. Monatsh. Math. 98(2), 145–155 (1984)
Rauzy, G.: Nombres algébriques et substitutions. Bull. Soc. Math. France 110(2), 147–178 (1982)
Schweiger, F.: Multidimensional continued fractions. Oxford Science Publications, Oxford University Press (2000)
Siegel, A., Thuswaldner, J.M.: Topological properties of Rauzy fractals. Mém. Soc. Math. Fr. (N.S.) (118), 140 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berthé, V., Bourdon, J., Jolivet, T., Siegel, A. (2013). Generating Discrete Planes with Substitutions. 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_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-40579-2_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40578-5
Online ISBN: 978-3-642-40579-2
eBook Packages: Computer ScienceComputer Science (R0)