Schröder's equation

Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function h(x), find the function Ψ(x) such that:

${\displaystyle \Psi (h(x))=s\Psi (x)~.}$

Schröder's equation is an eigenvalue equation for the composition operator C_h, which sends a function f(x) to f(h(x)).

If a is a fixed point of h(x), meaning h(a) = a, then either Ψ(a)=0 (or ∞) or s=1. Thus, provided Ψ(a) is finite and Ψ' (a) does not vanish or diverge, the eigenvalue s is given by s = h' (a).

For a = 0, if h is analytic on the unit disk, fixes 0, and 0 < |h′(0)| < 1, then Gabriel Koenigs showed in 1884 that there is an analytic (non-trivial) Ψ satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function.

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