Coincidence for substitutions of Pisot type

Marcy Barge; Beverly Diamond

Bulletin de la Société Mathématique de France (2002)

  • Volume: 130, Issue: 4, page 619-626
  • ISSN: 0037-9484

Abstract

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Let ϕ be a substitution of Pisot type on the alphabet 𝒜 = { 1 , 2 , ... , d } ; ϕ satisfies thestrong coincidence conditionif for every i , j ∈ 𝒜 , there are integers k , n such that ϕ n ( i ) and ϕ n ( j ) have the same k -th letter, and the prefixes of length k - 1 of ϕ n ( i ) and ϕ n ( j ) have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if d = 2 and provide a partial result for d ≥ 2 .

How to cite

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Barge, Marcy, and Diamond, Beverly. "Coincidence for substitutions of Pisot type." Bulletin de la Société Mathématique de France 130.4 (2002): 619-626. <http://eudml.org/doc/272474>.

@article{Barge2002,
abstract = {Let $\varphi $ be a substitution of Pisot type on the alphabet $\mathcal \{A\}=\lbrace 1, 2,\ldots , d\rbrace $; $\varphi $ satisfies thestrong coincidence conditionif for every $i, j \in \mathcal \{A\}$, there are integers $k, n$ such that $\varphi ^n(i)$ and $\varphi ^n(j)$ have the same $k$-th letter, and the prefixes of length $k-1$ of $\varphi ^n(i)$ and $\varphi ^n(j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d= 2$ and provide a partial result for $d \ge 2$.},
author = {Barge, Marcy, Diamond, Beverly},
journal = {Bulletin de la Société Mathématique de France},
keywords = {substitution; dynamical system; Pisot; coincidence conjecture; pure discrete spectrum},
language = {eng},
number = {4},
pages = {619-626},
publisher = {Société mathématique de France},
title = {Coincidence for substitutions of Pisot type},
url = {http://eudml.org/doc/272474},
volume = {130},
year = {2002},
}

TY - JOUR
AU - Barge, Marcy
AU - Diamond, Beverly
TI - Coincidence for substitutions of Pisot type
JO - Bulletin de la Société Mathématique de France
PY - 2002
PB - Société mathématique de France
VL - 130
IS - 4
SP - 619
EP - 626
AB - Let $\varphi $ be a substitution of Pisot type on the alphabet $\mathcal {A}=\lbrace 1, 2,\ldots , d\rbrace $; $\varphi $ satisfies thestrong coincidence conditionif for every $i, j \in \mathcal {A}$, there are integers $k, n$ such that $\varphi ^n(i)$ and $\varphi ^n(j)$ have the same $k$-th letter, and the prefixes of length $k-1$ of $\varphi ^n(i)$ and $\varphi ^n(j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d= 2$ and provide a partial result for $d \ge 2$.
LA - eng
KW - substitution; dynamical system; Pisot; coincidence conjecture; pure discrete spectrum
UR - http://eudml.org/doc/272474
ER -

References

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  1. [1] P. Arnoux & S. Ito – « Pisot substitutions and Rauzy fractals », Bull. Belg. Math. Soc. Simon Stevin8 (2001), p. 181–207. Zbl1007.37001MR1838930
  2. [2] P. Arnoux, S. Ito & Y. Sano – « Higher dimensional extensions of substitutions and their dual maps », J. Anal. Math.83 (2001), p. 183–206. Zbl0987.11013MR1828491
  3. [3] V. Berthé, S. Ferenczi, C. Mauduit & A. Siegel (éds.) – Introduction to Finite Automata and Substitutions Dynamical Systems, Lectures Notes in Mathematics, Springer-Verlag, to appear, http://iml. univ-mrs.fr/editions/preprint00/book/prebookdac.html. MR1970385
  4. [4] V. Canterini & A. Siegel – « Geometric representation of primitive substitutions of Pisot type », Trans. Amer. Math. Soc.353 (2001), p. 5121–5144. Zbl1142.37302MR1852097
  5. [5] F. Dekking – « The spectrum of dynamical systems arising from substitutions of constant length », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), p. 221–239. Zbl0348.54034MR461470
  6. [6] M. Hollander – « Linear numeration systems, finite β -expansions, and discrete spectrum of substitution dynamical systems », Thèse, University of Washington, 1996. MR2694876
  7. [7] M. Hollander & B. Solomyak – « Two-symbol Pisot substitutions have pure discrete spectrum », preprint to appear in Ergodic Theory and Dynamical Systems. Zbl1031.11010MR1972237
  8. [8] A. Livshits – « On the spectra of adic transformations of Markov compact sets », Uspekhi Mat. Nauk 42 (1987), p. 189–190, English translation: Russian Math. Surveys 42 (1987), p.222–223. Zbl0648.47004MR896889
  9. [9] —, « Sufficient conditions for weak mixing of substitutions and of stationary adic transformations », Mat. Zametki 44 (1988), p. 785–793, 862, English translation: Math. Notes 44 (1988), p.920-925. Zbl0668.28005MR983550
  10. [10] A. Siegel – « Représentation des systèmes dynamiques substitutifs non unimodulaires », preprint to appear in Ergodic Theory and Dynamical Systems. Zbl1052.37009
  11. [11] —, « Représentation géométrique, combinatoire et arithmétique des substitutions de type Pisot », Thèse, Université de la Méditerranée, 2000. 

Citations in EuDML Documents

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  1. Anne Siegel, Système dynamique à spectre discret et pavage périodique associé à une substitution
  2. Hiromi Ei, Shunji Ito, Hui Rao, Atomic surfaces, tilings and coincidences II. Reducible case
  3. Jörg M. Thuswaldner, Unimodular Pisot substitutions and their associated tiles
  4. Pierre Arnoux, Valérie Berthé, Arnaud Hilion, Anne Siegel, Fractal representation of the attractive lamination of an automorphism of the free group
  5. Valérie Berthé, Hiromi Ei, Shunji Ito, Hui Rao, On substitution invariant Sturmian words: an application of Rauzy fractals
  6. Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner, Dynamical directions in numeration

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