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Connectedness of fractals associated with Arnoux–Rauzy substitutions

Published online by Cambridge University Press:  27 May 2014

Valérie Berthé ,
Timo Jolivet  and
Anne Siegel
Valérie Berthé
Affiliation:
LIAFA, CNRS, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France.. [email protected]
Timo Jolivet
Affiliation:
LIAFA, CNRS, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France.. [email protected] Department of Mathematics, University of Turku 20014, Finland.
Anne Siegel
Affiliation:
INRIA, Centre Rennes-Bretagne Atlantique, Dyliss, Rennes, France.
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Abstract

Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux–Rauzy substitutions.

Keywords

Rauzy fractalsArnoux–Rauzy substitutionsdiscrete planes
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 48 , Issue 3: Special issue in the honor of the 14th “Journées montoises d’informatique théorique”. I. , July 2014 , pp. 249 - 266
Copyright
© EDP Sciences 2014

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References

Adamczewski, B., Frougny, C., Siegel, A. and Steiner, W., Rational numbers with purely periodic β-expansion. Bull. London Math. Soc. 42 (2010) 538–552. Google Scholar
Adler, R.L., Symbolic dynamics and Markov partitions. Bull. Amer. Math. Soc. (N.S.) 35 (1998) 1–56. Google Scholar
Akiyama, S. and Gjini, N., Connectedness of number theoretic tilings. Discrete Math. Theor. Comput. Sci. 7 (2005) 269–312 (electronic). Google Scholar
Akiyama, S., Barat, G., Berthé, V. and Siegel, A., Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. Monatsh. Math. 155 (2008) 377–419. Google Scholar
Arnoux, P. and Rauzy, G., Représentation géométrique de suites de complexit*error*é2n + 1. Bull. Soc. Math. France 119 (1991) 199–215. Google Scholar
Arnoux, P., Berthé, V., Fernique, T. and Jamet, D., Functional stepped surfaces, flips, and generalized substitutions. Theoret. Comput. Sci. 380 (2007) 251–265. Google Scholar
Arnoux, P., Berthé, V. and Ito, S., Discrete planes, Z2-actions, Jacobi-Perron algorithm and substitutions. Ann. Inst. Fourier 52 (2002) 305–349. Google Scholar
Arnoux, P., Berthé, V. and Siegel, A., Two-dimensional iterated morphisms and discrete planes. Theoret. Comput. Sci. 319 (2004) 145–176. Google Scholar
Arnoux, P. and Ito, S., Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8 (2001) 181–207. Google Scholar
Barge, M. and Kwapisz, J., Geometric theory of unimodular Pisot substitutions. Amer. J. Math. 128 (2006) 1219–1282. Google Scholar
Barge, M., Diamond, B. and Swanson, R., The branch locus for one-dimensional Pisot tiling spaces. Fund. Math. 204 (2009) 215–240. Google Scholar
Barge, M., Štimac, S. and Williams, R.F., Pure discrete spectrum in substitution tiling spaces. Discrete Contin. Dyn. Syst. 33 (2013) 579–597. Google Scholar
Berthé, V., Frettlöh, D., and Sirvent, V., Selfdual substitutions in dimension one, European J. Combin. 33 (2012) 981–1000. Google Scholar
V. Berthé and M. Rigo, Combinatorics, automata and number theory, Encyclopedia of Mathematics and its Applications, vol. 135. Cambridge University Press (2010).
V. Berthé, S. Ferenczi and L.Q. Zamboni, Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly, Algebraic and topological dynamics, Contemp. Math., vol. 385. Amer. Math. Soc. Providence, RI (2005) 333–364.
Berthé, V., Jolivet, T. and Siegel, A., Substitutive Arnoux-Rauzy sequences have pure discrete spectrum. Unif. Distrib. Theory 7 (2012) 173–197. Google Scholar
Berthé, V., Lacasse, A., Paquin, G. and Provençal, X., A study of Jacobi–Perron boundary words for the generation of discrete planes. Theoret. Comput. Sci. 502 (2013) 118–142. Google Scholar
Bowen, R., Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978) 130–132. Google Scholar
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, revised ed., Lect. Notes Math., vol. 470. With a preface by David Ruelle, edited by Jean–René Chazottes. Springer-Verlag, Berlin (2008).
Canterini, V., Connectedness of geometric representation of substitutions of Pisot type. Bull. Belg. Math. Soc. Simon Stevin 10 (2003) 77–89. Google Scholar
Canterini, V. and Siegel, A., Geometric representation of substitutions of Pisot type. Trans. Amer. Math. Soc. 353 (2001) 5121–5144. Google Scholar
Cassaigne, J. and Chekhova, N., Fonctions de récurrence des suites d’Arnoux-Rauzy et réponse à une question de Morse et Hedlund. Ann. Inst. Fourier Grenoble 56 (2006) 2249–2270. Google Scholar
Cassaigne, J., Ferenczi, S. and Messaoudi, A., Weak mixing and eigenvalues for Arnoux-Rauzy sequences. Ann. Inst. Fourier 58 (2008) 1983–2005. Google Scholar
Cassaigne, J., Ferenczi, S. and Zamboni, L.Q., Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265–1276. Google Scholar
Ei, H. and Ito, S., Decomposition theorem on invertible substitutions. Osaka J. Math. 35 (1998) 821–834. Google Scholar
Fernique, T., Multidimensional Sturmian sequences and generalized substitutions. Internat. J. Found. Comput. Sci. 17 (2006) 575–599. Google Scholar
Fernique, T., Generation and recognition of digital planes using multi-dimensional continued fractions. Pattern Recognition 42 (2009) 2229–2238. Google Scholar
Gazeau, J.-P. and Verger–Gaugry, J.-L., Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. J. Théor. Nombres Bordeaux 16 (2004) 125–149. Google Scholar
Hubert, P. and Messaoudi, A., Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals. Acta Arith. 124 (2006) 1–15. Google Scholar
Ito, S. and Ohtsuki, M., Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16 (1993) 441–472. Google Scholar
Ito, S. and Ohtsuki, M., Parallelogram tilings and Jacobi-Perron algorithm. Tokyo J. Math. 17 (1994) 33–58. Google Scholar
Ito, S. and Rao, H., Atomic surfaces, tilings and coincidence. I. Irreducible case. Israel J. Math. 153 (2006) 129–155. Google Scholar
D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995).
M. Lothaire, Combinatorics on words, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1997).
Messaoudi, A., Frontière du fractal de Rauzy et système de numération complexe. Acta Arith. 95 (2000) 195–224. Google Scholar
Morse, M. and Hedlund, G.A., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1–42. Google Scholar
Praggastis, B., Numeration systems and Markov partitions from self-similar tilings. Trans. Amer. Math. Soc. 351 (1999) 3315–3349. Google Scholar
N.P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lect. Notes Math., vol. 1794. Springer-Verlag, Berlin (2002).
M. Queffélec, Substitution dynamical systems-spectral analysis, second edition, Lect. Notes Math., vol. 1294. Springer-Verlag, Berlin (2010).
Rauzy, G., Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982) 147–178. Google Scholar
J.-P. Reveillès, Géométrie discrète, calculs en nombres entiers et algorithmes, Ph.D. thesis. Université Louis Pasteur, Strasbourg (1991).
A. Siegel, Représentations géométrique, combinatoire et arithmétique des systèmes substitutifs de type pisot, Ph.D. thesis. Université de la Méditerranée (2000).
A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractal. Mém. Soc. Math. Fr. To appear (2010).
Tan, B., Wen, Z.-X. and Zhang, Y., The structure of invertible substitutions on a three-letter alphabet. Adv. in Appl. Math. 32 (2004) 736–753. Google Scholar
W. Thurston, Groups, tilings, and finite state automata. AMS Colloquium lecture notes. Unpublished manuscript (1989).