Here we make some set theoretic statements, that, from my understanding are true. If any of the claims are false, please clarify the situation for me with a counter-example.
Let $f: A \to A$ be a bijective transformation.
Define, for each $x \in A$ the set $A_x =\{f^n(x) \mid \text{ integer } n \ge 0\}$.
If we call each set $A_x$ a filament, then the theory/notation found in
$\quad$ Generating the blocks of a partition from a family of filaments
is applicable here.
Before presenting the main theorem, we give an example of a bijective transformation on a set.
Recall for $n \ge 0$ we have the sets ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$.
Let $I$ be a set with a function
$\quad \displaystyle \tau: I \to \{\mathbb {Z} /n\mathbb {Z} \mid n \ge 0\}$
The mapping
$\quad m_{i} \mapsto m_i + 1 \text{ where } m_i \in \tau(i)$
defines a bijective tranformation on the (disjoint) union
$\quad {\displaystyle \bigsqcup _{i\in I}\,\tau(i)}$
We call such a transformation canonical.
Theorem: Every bijective tranformation on $A$, is, up to a (set) isomorphism, equivalent to a canonical transformation.
My work
I've been working towards describing a 'canonical picture' for the proof construction behind the Schröder-Bernstein theorem; the setting here is a simpler instance for the constructive analysis.
If the above passes scrutiny, I plan to provide the (canonical) proof detail for the Schröder-Bernstein theorem.
If the above theory/program has been already been worked out, any references/links would be appreciated.
Here is an example (see comment).
Let $A = \{1,2,3,4\}$ and
$\quad f(1) = 2, f(2) = 1, f(3) = 4, f(4) = 3$
Let
$\quad \displaystyle \tau: \{1,2\} \to \{\mathbb {Z} /2\mathbb {Z}\}$
be the constant mapping to the set $\mathbb {Z} /2\mathbb {Z} = \{0,1\}$ (the modulo 2 residue number system).
The canonical transformation representing $f$ operates on
$\quad \displaystyle \bigsqcup _{i\in \{1,2\}}\,\tau(i) = \bigsqcup _{i\in \{1,2\}} \{0,1\}_i$
by mapping $m_i \in \{0,1\}_i$ to $m_i + 1 \in \{0,1\}_i$ (a simple transposition, $0 \leftrightarrow 1$).
The trick to proving your theorem is to define $a \sim b$ if and only if there is some $n$ such that $f^n(a) = b$ or $f^n(b) = a$. This is an equivalence relation and the equivalence classes are the loops and $\Bbb Z$ images you are looking for.