Definition
Given that $f:\Bbb R\to\Bbb R$ is a real-valued function. The iteration of $f$, say $f^n$, is defined here:
- $f^0(x)=x$
- $f^n(x)=f\left(f^{n-1}(x)\right)$ for any positive integer $n$.
For example, $f(x)=x+c\implies f^{n}(x)=x+nc$, and $f(x)=2x\implies f^{n}(x)=2^nx$.
Question
Supposing that $f(x)$ is well-behaved enough, how can we explore the asymptotic value of $f^n(x)$?
A Concrete Example
For example, $f(x)=\sin x$, we can (very toughly to me) obtain that $\lim_{n\to\infty}\sqrt n\,f^n(x)=\sqrt 3$, therefore $f^n(x)=\sqrt{\frac 3n}+o(n^{-1/2})$ Can we obtain more details about $f^n(x)$?
Thoughts
Maybe we can suppose that $f(x)$ is analytic (at $x=0$), therefore we can rewrite $f(x)=\sum_{n\ge0}\alpha_nx^n$. Now can we obtain the details about the asymptotic value of $f^n(x)$?