Centralizers of a function $f$, specify $g$ such that $f\circ g=g \circ f$

Question

Let $f:R^n \to R^n$. I am looking for references on the description of centralizers of $f$, namely functions $g$ such that $f\circ g = g\circ f$.

Obviously $f^n$ (meaning the $n$-th iterate of $f$ with itself) is a candidate. What is $f^\infty$? What is the derivative of $f^\infty$?

For a matrix the infinite power question is settled.

For complex rational functions some results are known.

Answer 1

Presumably by $f^\infty(x)$ you mean $\lim_{n \to \infty} f^n(x)$. Usually this will not exist. If it does exist (for a particular $x$), and $f$ is continuous, then it is a fixed point of $f$. As a function of $x$, it might not be differentiable, or even continuous: an exception is in the case of an asymptotically stable fixed point, where $f^\infty$ would be locally constant.