Unimodular Pisot substitutions and their associated tiles

Thuswaldner, Jörg M.

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 487-536.

Soit $\sigma$ une substitution de Pisot unimodulaire sur un alphabet à $d$ lettres et soient $X_1,...,X_d$ les fractales de Rauzy associées. Dans le présent article, nous souhaitons étudier les frontières $\partial X_i$ ($1\le i\le d$) de ces fractales. Dans ce but, nous définissons un graphe, appelé graphe de contact de $\sigma$ et noté $𝒞$. Si $\sigma$ satisfait une condition combinatoire appelée condition de super coïncidence, le graphe de contact peut être utilisé pour établir un système auto-affine dirigé par un graphe dont les attracteurs sont des morceaux des frontières $\partial X_1,...,\partial X_d$. De ce système dirigé par un graphe, nous déduisons une formule simple pour la dimension fractale de $\partial X_i$, dans laquelle les valeurs propres de la matrice d’adjacence de $𝒞$ interviennent.

Un avantage du graphe de contact est sa structure relativement simple, ce qui rend possible sa construction immédiate pour une grande classe de substitutions. Dans cet article, nous construisons explicitement le graphe de contact pour une classe de substitutions de Pisot qui sont reliées aux $\beta$-développements par rapport à des unités Pisot cubiques. En particulier, nous considérons des substitutions de la forme

$$\sigma \left(1\right)=\underset{b\text{fois}}{\underbrace{1...1}}2,\sigma \left(2\right)=\underset{a\text{fois}}{\underbrace{1...1}}3,\sigma \left(3\right)=1$$

où $b\ge a\ge 1$. Il est bien connu que ces substitutions satisfont la condition de super coïncidence mentionnée plus haut. Donc nous pouvons donner une formule explicite pour la dimension fractale des frontìeres des fractales de Rauzy associées à ces substitutions.

Let $\sigma$ 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 $\partial X_i$ ($1\le i\le d$) of these fractals. To this matter we define a certain graph, the so-called contact graph $𝒞$ 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 $\partial X_1,...,\partial X_d$. 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 $𝒞$ 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

$$\sigma \left(1\right)=\underset{b\text{times}}{\underbrace{1...1}}2,\sigma \left(2\right)=\underset{a\text{times}}{\underbrace{1...1}}3,\sigma \left(3\right)=1$$

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.

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