Coincidence for substitutions of Pisot type

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

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Abstract

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Let $\varphi$ be a substitution of Pisot type on the alphabet $𝒜=\left\{1,2,...,d\right\}$; $\varphi$ satisfies thestrong coincidence conditionif for every $i,j\in 𝒜$, there are integers $k,n$ such that ${\varphi }^{n}\left(i\right)$ and ${\varphi }^{n}\left(j\right)$ have the same $k$-th letter, and the prefixes of length $k-1$ of ${\varphi }^{n}\left(i\right)$ and ${\varphi }^{n}\left(j\right)$ 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 $d\ge 2$.

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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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
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