Everybody who ever studied special relativity or hyperbolic trig knows this function
$$\tanh(Ax)$$
for real $0 < A < 1$
$\tanh(A x)$ has a nice attracting fixpoint at $0$ and it is asymptotic to a positive constant ($1$) for large positive $x$. It is also strictly increasing.
So this is very well suited to do complex dynamics on it.
So we consider $r$ iterations :
$$\tanh^{[r]}(A x)$$
for $r > 0$.
$$\tanh^{[0]}(A x) = x$$ $$\tanh^{[1]}(A x) = \tanh(A x)$$ $$\tanh^{[r+1]}(A x) = \tanh(A \tanh^{[r]}(A x))$$
Notice for every such $r$ (and fixed $A$)
$$\lim_{x \to +\infty} \tanh^{[r]}(A x) = \lim_{x \to 1/A} \tanh^{[r-1]}(A x) = C_r$$
!!Keep in mind!!: $$\tanh^{[r-1]}(1) \neq \lim_{x \to 1/A} \tanh^{[r-1]}(A x)$$
(because $A$ belongs to the part of the function that we iterate.)
Where $C$ is a constant depending on $r$.
I want to understand this for $0<A<1$.
Let us agree to use koenigs function around the fixpoint $0$ to generate the iterations or anything equivalent to that.
Then how does $C_r$ behave asymptotically ?
I added the tag tetration because it is a rational function of $\exp(x)$ afterall, hope that is ok.
See here : https://en.wikipedia.org/wiki/Koenigs_function
See also this here what is essentially equal to koenigs, but might converge at at different speed :
Theorem about fractional iterated functions : is it true, and if yes has it already been discovered?
$$f^k(x) = \lim_{n\rightarrow+\infty}(f^{-n}((f^n(x)-\tau)f'(\tau)^k +\tau))$$