Schröder's equation

Summary

Schröder's equation asks for a function $\Psi$ with $\Psi(f(x)) = \lambda \Psi(x)$, where the map $f$ and the constant $\lambda$ are given. A solution is a change of coordinates in which $f$ becomes multiplication by $\lambda$. It is one of the basic iterative functional equations: once it is solved, every iterate of $f$ has the closed form $f^n = \Psi^{-1} \circ (\lambda^n \Psi)$. Near a fixed point of a holomorphic map, the natural choice of $\lambda$ is the derivative of $f$ at that point.