Schröder's equation

Schröder's equation,

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

If a is a

Functional significance

For a = 0, if h is analytic on the unit disk, fixes 0, and 0 < |h′(0)| < 1, then

Moreover, for the velocity,[5] β(x) = Ψ/Ψ′,   Julia's equation,   β(f(x)) = f′(x)β(x), holds.

The n-th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue sⁿ, instead. In the same vein, for an invertible solution Ψ(x) of Schröder's equation, the (non-invertible) function Ψ(x) k(log Ψ(x)) is also a solution, for any periodic function k(x) with period log(s). All solutions of Schröder's equation are related in this manner.

Solutions

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).^[6][7]

In the case of a superattracting fixed point, |h′(a)| = 0, Schröder's equation is unwieldy, and had best be transformed to Böttcher's equation.^[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

It is used to analyse discrete dynamical systems by finding a new coordinate system in which the system (orbit) generated by h(x) looks simpler, a mere dilation.

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.

That is, h(x) = Ψ⁻¹(s Ψ(x)) ≡ h₁(x).

In general, all of its functional iterates (its regular iteration group, see iterated function) are provided by the orbit

for t real — not necessarily positive or integer. (Thus a full

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 instance, the functional square root is h_1/2(x) = Ψ⁻¹(s^1/2 Ψ(x)), so that h_1/2(h_1/2(x)) = h(x), and so on.

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),

Ψ(x) = (arcsin √x)², s = 4, and hence h_t(x) = sin²(2^t arcsin √x).

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

Ψ(x) = −1/2ln(1 − 2x), and hence h_t(x) = −1/2((1 − 2x)^2t − 1).

Likewise, for the Beverton–Holt model, h(x) = x/(2 − x), one readily finds^[10] Ψ(x) = x/(1 − x), so that^[13]

${\displaystyle h_{t}(x)=\Psi ^{-1}{\big (}2^{-t}\Psi (x){\big )}={\frac {x}{2^{t}+x(1-2^{t})}}.}$

See also

• Böttcher's equation

References