# Schröder's equation

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**Schröder's equation**,^{[1]}^{[2]}^{[3]} named after Ernst Schröder, is a functional equation with one independent variable: given the function *h*(*x*), find the function *Ψ*(*x*) such that:

Schröder's equation is an eigenvalue equation for the composition operator *C*_{h}, which sends a function *f*(*x*) to *f*(*h*(*x*)).

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

## Functional significance

For *a* = 0, if Template:Mvar is analytic on the unit disk, fixes 0, and 0 < |*h*′(0)| < 1, then Koenigs showed in 1884 that there is an analytic (non-trivial) Template:Mvar 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.

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*). It is also used in studies of turbulence, as well as the renormalization group.^{[4]}^{[5]}

An equivalent transpose form of Schröder's equation for the inverse *Φ*=*Ψ*^{−1} of Schröder's conjugacy function is *h*(*Φ*(*y*)) = *Φ*(*sy*). The change of variables *α*(*x*)=log(*Ψ*(*x*))/log(*s*) (the Abel function) further converts Schröder's equation to the older Abel equation, *α*(*h*(*x*)) = *α*(*x*)+1. Similarly, the change of variables Ψ(*x*) = log(*φ*(*x*)) converts Schröder's equation to Böttcher's equation, *φ*(*h*(*x*))=(*φ*(*x*))^{s}. 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 *s*^{n}, 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 Template:Mvar 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 Szekeres.^{[9]} Several of the solutions are furnished in terms of asymptotic series, cf. Carleman matrix.

## Applications

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*)) ≡ *h*_{1}(*x*).

In general, * all of its functional iterates* (its

*regular iteration group*, cf. iterated function) are provided by the

**orbit**

for Template:Mvar real — not necessarily positive or integer. (Thus a full continuous group.)
The set of *h*_{n}(*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 instance, the functional square root is *h*_{½}(*x*) = *Ψ*^{−1} (*s*^{½} *Ψ*(*x*) ), so that *h*_{½}( *h*_{½}(*x*) ) = *h* (*x*), and so on.

For example,^{[11]} special cases of the logistic map such as the chaotic case *h*(*x*)=4*x*(1−*x*) were already worked out by Schröder in his original paper^{[1]} (cf. p. 306),

*Ψ*(*x*) = arcsin²(√*x*),*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*) = 2*x*(1−*x*), yields

*Ψ*(*x*) = −½ln(1−2*x*), and hence*h*_{t}(*x*) = −½((1−2*x*)^{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]}

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## References

- ↑
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- ↑
^{10.0}^{10.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }} - ↑ Curtright, T.L. Evolution surfaces and Schröder functional methods.
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Skellam, J.G. (1951). “Random dispersal in theoretical populations”,
*Biometrika***38**196−218, eqns (41,42)