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Geometrical representation of subshifts for primitive substitutions

Part of: Topological dynamics

Published online by Cambridge University Press:  03 November 2023

PAUL MERCAT
PAUL MERCAT*
Affiliation:
I2M, Aix-Marseille University, 3 Place Victor Hugo, 13331 Marseille, France
*
e-mail: paul.mercat@univ-amu.fr
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Abstract

For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.

Keywords

substitution subshiftsRauzy fractalsfinite extension of torus translationeigenvalues of subshifts

MSC classification

Primary: 37B10: Symbolic dynamics

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 7 , July 2024 , pp. 1885 - 1912
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Arnoux, P., Bernat, J. and Bressaud, X.. Geometrical models for substitutions. Exp. Math. 20(1) (2011), 97–127.CrossRefGoogle Scholar
Arnoux, P. and Ito, S.. Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. 8 (2001), 181–207.Google Scholar
Akiyama, S. and Mercat, P.. Yet another characterization of the Pisot substitution conjecture. Substitution and Tiling Dynamics: Introduction to Self-Inducing Structures. Eds. Akiyama, S. and Arnoux, P.. Springer, Cham, 2020, Ch. 8.CrossRefGoogle Scholar
Andrieu, M.. A Rauzy fractal unbounded in all directions of the plane. C. R. Acad. Sci. 359(4) (2021), 399–407.Google Scholar
Berthé, V. and Siegel, A.. Tilings associated with $\beta$ -numeration and substitutions. Integers 5(3) (2005), A02.Google Scholar
Canterini, V. and Siegel, A.. Automate des préfixes-suffixes associé à une substitution primitive. J. Théor. Nombres Bordeaux 13(2) (2001), 353–369.CrossRefGoogle Scholar
Dekking, F. M.. The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitsh. Verw. Geb. 41 (1978), 221–239.CrossRefGoogle Scholar
Petite, S. and Durand, F.. Conjugacy of unimodular Pisot substitution subshifts to domain exchanges. Preprint, 2020, arXiv:1408.2110. https://hal.archives-ouvertes.fr/hal-01053723v2/document.Google Scholar
Durand, F.. A characterization of substitutive sequences using returns words. Discrete Math. 179 (1998), 89–101.CrossRefGoogle Scholar
Ferenczi, S., Mauduit, C. and Nogueira, A.. Substitution dynamical systems: algebraic characterization of eigenvalues. Ann. Sci. Éc. Norm. Supér. (4) 29(4) (1996), 519–533.CrossRefGoogle Scholar
Host, B.. Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6 (1986), 529–540.CrossRefGoogle Scholar
Mossé, B.. Reconnaissabilité des substitutions et complexité des suites automatiques. Bull. Soc. Math. France 124(2) (1996), 329–346.CrossRefGoogle Scholar
Pytheas Fogg, N. and Noûs, C.. Symbolic coding of linear complexity for generic translations on the torus, using continued fractions. Preprint, 2020, arXiv:2005.12229.Google Scholar
Queffélec, M.. Substitution Dynamical Systems – Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.CrossRefGoogle Scholar
Rauzy, G.. Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982), 147–178.CrossRefGoogle Scholar
Sirvent, V. F. and Wang, Y.. Self-affine tiling via substitution dynamical systems and Rauzy fractals. Pacific J. Math. 206(2) (2002), 465–485.CrossRefGoogle Scholar
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