I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$
that results by iterating inversion in a unit circle.
Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle
centered on $q_1$, and then in a unit circle centered on $q_2$,
etc. Here is an example:

The black path shows the trajectory of one point $z$, which
ends up inside the $q_5$ disk.
The blue vectors show the complete map's effect on a number of
(random) points.
The end result resembles an inversion in the $q_5$ circle,
but I imagine the plane is partitioned into regions that
behave similarly due to their relationship to the several circles of
inversion.
Is this a correct way to view this map?
And if so, is there a natural diagram to elucidate the partition?

I wonder if instead it may be better to view this map as an approximation to another map? For there is a continued-fraction representation of $f(z)$, as follows. The inversion of $z$ in a unit circle centered on $q$ can be expressed as $q + \frac{1}{z^* -q^*}$, where ${\cdot}^*$ represents the complex conjugate operation. Then iterating this (say, five times) leads to this expression: $$f(z) = q_5 + \frac{1} {q_4^*-q_5^* + \frac{1} {q_3-q_4+ \frac{1} {q_2^*-q_3^*+ \frac{1} {q_1-q_2+\frac{1}{z^*-q_1^*} } } } } $$

I realize I have not posed a sharp question. I am just seeking some approach that helps understand the composition of circle inversions, a bit far from my expertise. Thanks for pointers or ideas!