Suppose I have a discrete dynamical system given by: $$ X^{n+1} = f(X^{n}) \qquad X^0 =x , $$ where $f$ is some diffeomorphism on $\mathbb{R}^d$ and $x \in \mathbb{R}^d$. When does there exist a function $\tilde{f}:\mathbb{R}^d\rightarrow \mathbb{R}^d$ such that:
$$ Z^{n+1}= Z^{n} + \tilde{f}(Z^{n}) \qquad Z^0=x, $$
and (either of):
Not Ideally:
and for every $\epsilon>0$ there is some $n_{\epsilon}>0$ satisfying: $$ \|Z^n - X^n\|<\epsilon \qquad (\forall n\geq n_{\epsilon})? $$
Ideally:
Or it possible, the stronger condition holds: $X^n = Z^n$ for all large $n$?
It suffices to take $\tilde f(x) = f(x) - x$, which leads to $Z^n = X^n$ for all $n$.