Unimodular Pisot substitutions and their associated tiles

Jörg M. Thuswaldner[1]

  • [1] Institut für Mathematik und Angewandte Geometrie Abteilung für Mathematik und Statistik Montanuniversität Leoben Franz-Josef-Strasse 18 A-8700 LEOBEN, AUSTRIA

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 2, page 487-536
  • ISSN: 1246-7405

Abstract

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Let σ be a unimodular Pisot substitution over a d letter alphabet and let X 1 , ... , X d be the associated Rauzy fractals. In the present paper we want to investigate the boundaries ∂ X i ( 1 ≤ i ≤ d ) of these fractals. To this matter we define a certain graph, the so-called contact graph 𝒞 of σ . If σ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries ∂ X 1 , ... , ∂ X d . From this graph directed system we derive an easy formula for the fractal dimension of ∂ X i in which eigenvalues of the adjacency matrix of 𝒞 occur.An advantage of the contact graph is its relatively simple structure, which makes it possible to construct it for large classes of substitutions at once. In the present paper we construct the contact graph explicitly for a class of unimodular Pisot substitutions related to β -expansions with respect to cubic Pisot units. In particular, we deal with substitutions of the form σ ( 1 ) = 1 ... 1 ︸ b times 2 , σ ( 2 ) = 1 ... 1 ︸ a times 3 , σ ( 3 ) = 1 where b ≥ a ≥ 1 . It is well known that these substitutions satisfy the above mentioned super coincidence condition. Thus we can give an explicit formula for the fractal dimension of the boundaries of the Rauzy fractals related to these substitutions.

How to cite

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Thuswaldner, Jörg M.. "Unimodular Pisot substitutions and their associated tiles." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 487-536. <http://eudml.org/doc/249630>.

@article{Thuswaldner2006,
abstract = {Let $\sigma $ be a unimodular Pisot substitution over a $d$ letter alphabet and let $X_1,\ldots , X_d$ be the associated Rauzy fractals. In the present paper we want to investigate the boundaries $\partial X_i$ ($1\le i\le d$) of these fractals. To this matter we define a certain graph, the so-called contact graph$\mathcal\{C\}$ of $\sigma $. If $\sigma $ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $\partial X_1,\ldots , \partial X_d$. From this graph directed system we derive an easy formula for the fractal dimension of $\partial X_i$ in which eigenvalues of the adjacency matrix of $\mathcal\{C\}$ occur.An advantage of the contact graph is its relatively simple structure, which makes it possible to construct it for large classes of substitutions at once. In the present paper we construct the contact graph explicitly for a class of unimodular Pisot substitutions related to $\beta $-expansions with respect to cubic Pisot units. In particular, we deal with substitutions of the form\begin\{equation*\} \sigma (1) = \underbrace\{1\ldots 1\}\_\{b\,\hbox\{times\}\}2, \quad \sigma (2) =\underbrace\{1\ldots 1\}\_\{a\,\hbox\{times\}\}3, \quad \sigma (3) =1 \end\{equation*\}where $b\ge a\ge 1$. It is well known that these substitutions satisfy the above mentioned super coincidence condition. Thus we can give an explicit formula for the fractal dimension of the boundaries of the Rauzy fractals related to these substitutions.},
affiliation = {Institut für Mathematik und Angewandte Geometrie Abteilung für Mathematik und Statistik Montanuniversität Leoben Franz-Josef-Strasse 18 A-8700 LEOBEN, AUSTRIA},
author = {Thuswaldner, Jörg M.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {unimodular Pisot substitution; tiles; Pisot-Vijayaraghavan number; Rauzy fractals; contact graph},
language = {eng},
number = {2},
pages = {487-536},
publisher = {Université Bordeaux 1},
title = {Unimodular Pisot substitutions and their associated tiles},
url = {http://eudml.org/doc/249630},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Thuswaldner, Jörg M.
TI - Unimodular Pisot substitutions and their associated tiles
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 487
EP - 536
AB - Let $\sigma $ be a unimodular Pisot substitution over a $d$ letter alphabet and let $X_1,\ldots , X_d$ be the associated Rauzy fractals. In the present paper we want to investigate the boundaries $\partial X_i$ ($1\le i\le d$) of these fractals. To this matter we define a certain graph, the so-called contact graph$\mathcal{C}$ of $\sigma $. If $\sigma $ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $\partial X_1,\ldots , \partial X_d$. From this graph directed system we derive an easy formula for the fractal dimension of $\partial X_i$ in which eigenvalues of the adjacency matrix of $\mathcal{C}$ occur.An advantage of the contact graph is its relatively simple structure, which makes it possible to construct it for large classes of substitutions at once. In the present paper we construct the contact graph explicitly for a class of unimodular Pisot substitutions related to $\beta $-expansions with respect to cubic Pisot units. In particular, we deal with substitutions of the form\begin{equation*} \sigma (1) = \underbrace{1\ldots 1}_{b\,\hbox{times}}2, \quad \sigma (2) =\underbrace{1\ldots 1}_{a\,\hbox{times}}3, \quad \sigma (3) =1 \end{equation*}where $b\ge a\ge 1$. It is well known that these substitutions satisfy the above mentioned super coincidence condition. Thus we can give an explicit formula for the fractal dimension of the boundaries of the Rauzy fractals related to these substitutions.
LA - eng
KW - unimodular Pisot substitution; tiles; Pisot-Vijayaraghavan number; Rauzy fractals; contact graph
UR - http://eudml.org/doc/249630
ER -

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