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Im translating an article about Rauzy fractal and I ran into this sentence:

The Rauzy fractal has remarkable properties. Firstly, it is selfsimilar,
more exactly, it is divided into three pieces,
corresponding to the three letters, which are the
solutions of a graph-directed iterated function system.

I do not know what is a graph-directed iterated function system. Can any one help me? Iv searched the net but I didn't find anything useful.

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From the wikipedia article, you have the substition rules $1\to12$, $2\to13$, $3\to1$. We can write this in matrix form $$ \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} $$ where the first column indicate what $1$ is mapped to, the second what $2$ is mapped to, and so on. This makes a bit sense, the characteristic polynomial of this matrix is $t^3-t^2-t-1$, which is also mentioned in the article.

Now, this matrix can also be seen as the incidence matrix of a directed graph, with edges $1\to 1, 1\to 2, 2\to 1, 2\to 3$ and so on, so perhaps this is what is meant by directed-graph iterated function system.

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  • $\begingroup$ A reference is "Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure," Boore and Falconer. The setting is slightly more general, in that each vertex of the graph can have its own metric space. Note also the comment "We follow the convention already established in the literature, see [5] or [6], that Se maps in the opposite direction to the direction of the edge it is associated with in the graph." $\endgroup$
    – user25199
    Commented Nov 21, 2014 at 9:37

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