The book considers mappings $a, b, A,$ and $B$ where $A = a^{-1}, B = b^{-1}$.
It goes on to say that words represented by compositions of these maps (e.g. $abbA$) correspond to points.
I don't understand this part. The book never explains how to get from a word to a point.
Here's an example of where my thought process gets stuck:
The book considers as an example a word $W = aBB$ and asks for the corresponding fixed point. It gives the answer as just $WWWWWW.....$ (W repeating). Later in the book it considers $a, b, A$, and $B$ as complex matrices, so then to approximate the fixed point of $a$, you'd just take a big power of $a$. But it also says the fixed points are complex numbers and doesn't say how to get the numbers.
How do I get the actual points corresponding to words?
The goal of this question is to figure out the above so that I can figure out what to put in the fix[] array given in the pseudo-code on page 152.
To get from words to points, you need to pick a random complex number ("seed point") $z_0 \in \mathbb{C}$ and apply the word to it. But as long as you're plotting long words, the final picture is the same no matter what point you start with. (See p.132-133 of the book.)
The "fix" array contains the attractive fixed point of each of the Mobius transformations $a,b,A,B$. This is the point $z$ satisfying $f(z)=z$ and $|f'(z)|<1$. Since the transformations are of the form $(az+b)/(cz+d)$, the fixed points can be found by solving a quadratic equation.