Poincaré's functional equation

Summary

Poincaré's functional equation asks for a function $\Psi$ with $\Psi(\lambda z) = f(\Psi(z))$, where $f$ is a given holomorphic map with a fixed point at $0$ and $\lambda = f'(0)$. It is the inverse form of Schröder's equation: if $\Phi$ solves Schröder's equation and is invertible, then $\Psi = \Phi^{-1}$ solves Poincaré's equation. For a repelling fixed point of an entire map, the solution extends to an entire function on all of $\C$, the Poincaré function. Poincaré functions parametrise the dynamics of $f$ near the repelling point and are used to study Julia sets.