X-th iteration where x is a real number

[Natsugou]

A sequence \(a\) defined by a recurrence \(a_{n+1} = f(a_n)\) (i.e. a sequence satisfies \(a_i = a_j \implies a_{i+1} = a_{j+1}\)) is eventually periodic
\[
    0, 1, 2, 3, 4, 5, 3, 4, 5, \dots
\]
or injective.

I verified a function of real variable \(a\) such that satisfies \(a(x) = a(y) \implies a(x+u) = a(y+u)\) for all \(u > 0\) is characterized by a subgroup and a cut of \((\mathbb{R}, +)\) in this PDF. 
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I defined periods of \(a\) \(P(a) := \{g^{-1}h \in G \mid a(g) = a(h), g<h\}\), heights of \(a\) \(H(a) := \{h \in G \mid \exists p \in G_+, a(h) = a(hp)\}\).

For any such \(a\), \(\pm P_0(a) = -P(a) \sqcup \{0\} \sqcup P(a)\) and \((\mathbb{R}\backslash H(a), H(a))\) are a subgroup and a cut of \((\mathbb{R}, +)\).
Moreover, \(a(x) = a(y) \iff b(x) = b(y)\) where \(b\) is defined as
\[
    b(x) :=
    \begin{cases}
        (x, 1) & \text{if \(x \notin H(a)\)} \\
        (x+\pm P_0(a), 2) & \text{if \(x \in H(a)\)}.
    \end{cases}
\]

However, this gives us very little information because I did not define any structure on the codomain of \(a\).
Which should I consider the case the codomain of \(a\) is a topological space, or the domain of \(a\) is not an archimedean group?

[MphLee]

Hi, this is a very interesting contribute in the foundation of general semigorup-time dynamics/iteration. As such I move it from this section to the main one.