I would like to know whether the following statement or a weaker / similar version of it is true. Intuitively, it seems to me that it should be true but I'm unable to prove it. Does anyone know a theorem with a name that states something like the below? Thanks so much.
Let $f: [a,b] \rightarrow \mathbb{R}$ with $a, b \in \mathbb{R}$ be a differentiable function defined on some interval $[a,b]$ and let $c \in [a,b]$ be such that
- $f^n(c) \in [a,b]$ for all $n$, where $f^n$ denotes function iteration
- for all $d \in [a,b]$ for which there exists an $m \ge 1$ with $f^{m}(d) = d$ we have $f(d) = d$
Then $f^n(c)$ converges as $n$ converges to infinity.