Given a function $f\colon \Bbb Z\to\Bbb Z$. Assume that for some index set $I$, there are $a_i,b_i,c_i,d_i\in\Bbb Z$ such that
- $\Bbb Z=\bigcup_{i\in I}(a_i+b_i\Bbb N_0)$ and $\Bbb Z=\bigcup_{i\in I}(c_i+d_i\Bbb N_0)$
- For all $i\in I$, $n\in\Bbb N_0$, we have $f(a_i+b_in)=c_i+d_in$
I am looking for references dealing with the properties of trees and digraphs formed from $\{\,f^{\circ k}(x)\mid k\in\Bbb N_0\,\}$ for all $x\in \Bbb Z$.
For example:
$$f(x) = \begin{cases} \frac{3}{2}x, & \text{subdoman:2, 4, 6, 8,...Subrange: 3, 6, 9, 12...} \\ \frac{-3}{2}x+\frac{1}{2}, & \text{subdoman:1, 3, 5, 7,...Subrange:-1,-4,-7, -10...}\\ \frac{-1}{2}x+\frac{1}{2}, & \text{subdoman:-1, -3, -5,...Subrange:1, 2, 3, ...}\\ \frac{1}{2}x, & \text{subdoman:-2, -4, -6,...Subrange:-1, -2, -3,...} \end{cases}$$
Part of the tree is
And the digraph is
I have read the wikipedia article on iterated functions but it was more on fixed points. Appreciate any help.