I apologize in advance for the naive nature of the following questions. I am also thankful to suggestions for improving the direction of the questions instead of direct answers.
Let $f: \mathbb N \to \mathbb N$ be a map on the integers. Consider the dynamics of the iteration of this map, i.e., let $a \in \mathbb N$ be an initial value and consider the sequence of iterates $(a, f(a), f^2(a) = (f\circ f)(a), f^3(a),...)$. Natural questions that occur are:
1) What is the "cycle-structure" of the digraph associated to the dynamics of the map $f$ (i.e. the "phase portrait"), in particular: are there limit cycles and what is their length?
2) For which initial values the trajectory diverges?
Now my question is the following and a bit vague: In which way is it possible to answer "inverse questions" of 1) and 2): For example, is it trivial or difficult to find a map $f: \mathbb N \to \mathbb N$ with a limit cycle of prescribed length or even with a prescribed set of limit cycles with given lengths?(of course simply assigning single values is not meant here but an explicit formula for $f$ or possibly defining $f$ by cases, where the single cases are explicit formulas and every case covers infintely many numbers). What can we say about the class of functions such a map $f$ (fulfilling the requested conditions on the cycle-structure) belongs to? For example, is it possible to achieve this by a polynomial, rational function, etc of certain degree..
I also wonder about the analogous question for a map $f: X \to X$ on a finite set $X$. More concretely, let $X = \{0,...,n-1\}$. I've found a lot of papers about the structure of the functional digraph of special $f$, in particular $n$ being a prime number or a modulus with a primitive root and $f$ being some modular exponentiation. For finite $X$, a connected component of a functional digraph consists of a cycle (or loop) and possibly some trees ("tails") leading to the cycle. My question here is in the "modular setting": for what kind of functions $f$ (more specialized: which polynomials) can we expect the problem of completely describing the structure of the functional digraph to be tractable and for which $f$ out of reach, and for what reasons? And do these kind of investigations yield nontrivial results about number theory, not known before? (for example finding explicit primitive roots for primes of certain forms).
There is very little chance that your question has anything but an extremely general answer, for the following reasons.
Any digraph having the special property that there is exactly one outgoing edge at each vertex corresponds to a self-map of its vertex set. Let me call these "function" digraphs.
So, any function digraph with a countable infinity of vertices corresponds to a self-map $f : \mathbb{N} \to \mathbb{N}$. For example, you can take any countable number of connected function digraphs each with countably many vertices, and take their disjoint union. By this means you can produce examples with any countable number of fixed points, cycles of order 2, cycles of order 3, etc., any countable number of nonpreperiodic orbits, etc.
Similarly, any function digraph with $n$ vertices corresponds to a self-map of a set of cardinality. So you can get examples with any number of fixed points, cycles of order 2, of order 3, etc., as long as the sum of the cardinalities of the cycles is at most $n$.