I've looked into iterated functions for a bit more than a year (especially thanks to tetration), but there's still things I do not quite know about them, especially when online searching isn't really concluent.
Just for you know, an iterated function is the function $F(x)$ defined as :
$F(X)=f^{\circ X}(x_0)=\underbrace{f(f(f(...x_0)))}_{X~times}$
Where $F^{\circ 0}(x_0)=x_0$
Such iterated functions are pretty easy to calculate if they have a fixed point, if they converge to a finite value. Thanks to Schroeder's equation :
For any function $f$, if you find a function $Ψ$ such that $Ψ(f(x))-τ = λ(Ψ(x)-τ)$, then $f^{\circ X}(x_0)=Ψ^{-1}((Ψ(x_0)-τ)λ^X+τ)$ (There are restrictions about what inverse function to use, and which $x_0$ you can chose but it's not the topic)
($τ$ and $λ$) are constants
In this case, with convergent iterated functions, $Ψ(x)=\lim_{n\to+\infty}({f^{\circ n}(x)})$, $τ$ is the fixed point the function converges to (with $x_0$ as the starting number), and $λ=f'(τ)$
But sometimes, the function doesn't converge, it doesn't have a fixed point. Actually, the double-iterated function $f(f(x))$ has 2 fixed points, so the iterated function oscillates between 2 values that converges.
In that case, I thought about calculating 2 different iterations, one by calculating the iterations of $f(f(x_0))$, and anoter by calculating the iterations of the same thing, but taking $f(x_0)$ as the starting number (basically decomposing the oscillation by its upper part and lower part, turning it into 2 new iterated functions that converge), and then by combining them with sines and cosines, but it wasn't concluent.
So if any of you know a "oscillating" version of Schroeder's equation, I'd be glad to hear it!