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Unimodular Pisot substitutions and their associated tiles. (English) Zbl 1161.37016
Let $$\sigma$$ be a unimodular Pisot substitution over a $$d$$-letter alphabet and let $$X_1,\dots,X_d$$ be the associated Rauzy fractals. In this paper the author investigates the boundaries $$\partial X_i$$ ($$1\leq i\leq d$$) of these fractals. To this matter, he defines the contact graph $$\mathcal{C}$$ of $$\sigma$$. If $$\sigma$$ satisfies the super coincidence condition [see S. Ito and H. Rao, Isr. J. Math. 153, 129–155 (2006; Zbl 1143.37013)] – also sometimes called the geometric coincidence condition [see M. M. Barge and B. Diamond, Bull. Soc. Math. Fr. 130, 619–626 (2002; Zbl 1028.37008)] – the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $$\partial X_i$$ ($$1\leq i\leq d$$).
From this graph directed system the author derives a formula for the fractal dimension of $$\partial X_i$$ in which eigenvalues of the adjacency matrix of $$\mathcal{C}$$ occurs. The main advantage of the contact graph is its relatively easy shape. The author calculates the contact graph for the substitutions $$\sigma(1)=1^b 2$$, $$\sigma(2)=1^a 3$$ and $$\sigma(3)=1$$ with $${b\geq a\geq 1}$$. It turns out that it has roughly the same shape for each substitution from this class. The knowledge of the contact graphs of these substitutions enables us to establish an explicit formula for the Hausdorff dimension of the boundary of the associated Rauzy fractals.

##### MSC:
 37B10 Symbolic dynamics 11B85 Automata sequences 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28A80 Fractals 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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