---
tag: 3384ZV3PWM
kind: concept
name: Poincaré's functional equation
msc: [39B32, 30D05, 37F10]
specializes: [BCGBQ3HK26]
sources:
  - url: https://en.wikipedia.org/wiki/Schr%C3%B6der%27s_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|R6QVS6P21N]] at $0$ and $\lambda = f'(0)$.
It is the inverse form of [[Schröder's equation|VTJV7JVJEA]]: 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.

## Explanations

### L1 human

Schröder's equation looks for a coordinate in which the map $f$ becomes multiplication by $\lambda$.
Poincaré's equation looks at the same change of coordinates in the other direction: it maps the simple picture, multiplication by $\lambda$, onto the complicated one.
For $f(w) = w^2 + 2w$ and $\lambda = 2$, the function $\Psi(z) = e^z - 1$ works, since $f(e^z - 1) = e^{2z} - 1 = \Psi(2z)$.
For the logistic map $f(w) = 4w(1 - w)$ and $\lambda = 4$, the function $\Psi(z) = \sin^2 \sqrt{z}$ works, by the double-angle formula.