Construction of Rauzy Fractals with substitutions without a fixed point

Question

The formal definition of a Rauzy fractal can be found at the beginning of this paper

Using Sage-math-cloud, I can generate Rauzy fractals of substitutions that I choose. Should I choose the substitution $\sigma: 1 \to 21, \; 2\to 31, \; 3 \to 1$, the Sage spits out a fractal which looks exactly like the Rauzy fractal for the Tribonacci substitution, but merely shifted over.

My question is, how can $\sigma$ have a Rauzy fractal representation at all if it doesn't have a fixed point? By definition, we use the fixed point $w$ to construct our 'staircase' of basis vectors. Do they mean the $\textit{periodic}$ point? That would make some sense since every primitive substitution has a periodic point.

Answer 1

I have only seen the Rauzy fractal defined using a fixed point of a primitive substitution. An example is in this paper by Sirvent and Wang, which I find to be a particularly concise and relatively straightforward introduction to the Rauzy fractal. I previously referred to the paper in this answer and summarized the basic idea of the construction.

However, it's easy to see why you should be able to generate a Rauzy-like image using a substitution with a periodic point. Let's consider your substitution, for example: $$\sigma: 1 \to 21, \; 2\to 31, \; 3 \to 1.$$ If we start with the simple string "$1$" and iteratively perform the substitution, we obtain the sequence:

1    
21    
3121    
1213121    
2131211213121    
312112131212131211213121    
12131212131211213121312112131212131211213121
213121121312131211213121213121121312112131212131211213121312112131212131211213121
31211213121213121121312112131212131211213121312112131212131211213121213121121312131211213121213121121312112131212131211213121312112131212131211213121
1213121213121121312131211213121213121121312121312112131213121121312121312112131211213121213121121312131211213121213121121312131211213121213121121312112131212131211213121312112131212131211213121213121121312131211213121213121121312112131212131211213121312112131212131211213121

We see clearly the convergence to a sequence of period three.

Now, consider the following substitution: $$\sigma^3: 1 \to 1213121, \; 2\to 213121, \; 3 \to 3121.$$ This was obtained by applying the original substitution three times to the three initial character strings "$1$", "$2$", and "$3$". Now $\sigma^3$ has a fixed point which is exactly one of the periodic points of your original sequence. The resulting Rauzy fractal is probably what you see in your image. Alternatively, the other two strings lead to images that geometrically similar to this one. I guess this "uniqueness" is why we prefer a fixed point to a periodic one.