[Question] Classifying dynamical system by connected components

[Natsugou]

I don't know category theory nor colimit, but may I ask you questions about the definitions of \(\mathrm{gen}(X)\) and \(\mathrm{ete}(X)\)?
Is the definition of \(\mathrm{gen}(X)\) \(\mathrm{gen}(X) := \{x \in X \mid \forall y \in X, f(y) \neq x\}\)?
And which one of the three is the definition of \(\mathrm{ete}(X)\), where \(X' = \{x \in X \mid \exists! y \in X, f(y) = x\}\),
\[
  \begin{array}{ccccl}
    p: & X' & \to    & X' & \\
      & x  & \mapsto & y  & \mathrm{s.t.}\; f(y) = x,
  \end{array}
\]
\(X'' = \{x \in X' \mid \forall n \in \mathbb{N}, p^n(x) \in X'\}\), \((X'', p)\) is a \(\mathbb{N}\mathbf{Set}\),
and \(A\) is the set of decomposed connected dynamical sub-systems of \((X'', p)\)? (i.e. \((X'', p) = \bigsqcup_{(e, p) \in A}(e, p)\).)

1. \(\mathrm{ete}(X) := A\).
2. \(\mathrm{ete}(X) := \{(e, p) \in A \mid (e, p) \text{is generated}\}\).
3. \(\mathrm{ete}(X) := \{(e, p) \in A \mid (e, p) \text{is infinite}\}\).
           

(edit 2025-09-22) I noticed that my definition of \(\mathrm{ete}(X)\) written above doesn't work well when \(X\) is a dynamical system like the infinite complete binary tree. What is \(\mathrm{ete}(X)\) of such \(X\)?