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.



What is "graph-directed iterated function"?
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.


