Say we have a function $f(x)$ such that $f(0)\neq0$ and construct its iterates at zero e.g. $f^3(0)=f(f(f(0)))$. Let it also be a one-one function such that it has a unique inverse so the $f^{-1}(0)$ is well defined.
Is there a way to reconstruct the function $f(x)$ from all the iterates $f^N(0)$ where N can be any positive or negative integer?
As an example, if $f(x)=x+a$ then $f^n(0)=na$. From the values $\{..,-2a,-a,0,a,2a,3a,...\}$ it is easy to reconstruct the original function.
Is there such a series? e.g.
$$f(x) = f^1(0) +...$$
(Or at least construct it up to the symmetry:
$$ f(x)\rightarrow h^{-1}f(h(x))$$ where $h$ is any function such that $h(0)=0$ e.g. $h(x)=x^2$)
Even for your example, there is no particular reason to assume that $f(x)=x+a$ from the iterates, other than simplicity. The function could be doing all sorts of things between $k\cdot a$ and $(k+1)\cdot a$ (so long as it preserves injectivity).