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.

Equations such as Schröder's are suitable to encoding self-similarity, and have thus been extensively utilized in studies of nonlinear dynamics (often referred to colloquially as chaos theory).

Schröder's equation was solved analytically if a is an attracting (but not superattracting) fixed point, that is 0 < |h′(a)| < 1 by Gabriel Koenigs (1884).

[8] There are a good number of particular solutions dating back to Schröder's original 1870 paper.

[1] The series expansion around a fixed point and the relevant convergence properties of the solution for the resulting orbit and its analyticity properties are cogently summarized by Szekeres.

More specifically, a system for which a discrete unit time step amounts to x → h(x), can have its smooth orbit (or flow) reconstructed from the solution of the above Schröder's equation, its conjugacy equation.

The set of hn(x), i.e., of all positive integer iterates of h(x) (semigroup) is called the splinter (or Picard sequence) of h(x).

However, all iterates (fractional, infinitesimal, or negative) of h(x) are likewise specified through the coordinate transformation Ψ(x) determined to solve Schröder's equation: a holographic continuous interpolation of the initial discrete recursion x → h(x) has been constructed;[10] in effect, the entire orbit.

For example,[11] special cases of the logistic map such as the chaotic case h(x) = 4x(1 − x) were already worked out by Schröder in his original article[1] (p. 306), In fact, this solution is seen to result as motion dictated by a sequence of switchback potentials,[12] V(x) ∝ x(x − 1) (nπ + arcsin √x)2, a generic feature of continuous iterates effected by Schröder's equation.

A nonchaotic case he also illustrated with his method, h(x) = 2x(1 − x), yields Likewise, for the Beverton–Holt model, h(x) = x/(2 − x), one readily finds[10] Ψ(x) = x/(1 − x), so that[13]