Abel equation

The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form

f (h (x)) = h (x + 1)

or

α (f (x)) = α (x) + 1

.

The forms are equivalent when $α$ is invertible. $h$ or $α$ control the iteration of $f$ .

Equivalence

The second equation can be written

α^{- 1} (α (f (x))) = α^{- 1} (α (x) + 1) .

Taking $x = α -1 (y)$ , the equation can be written

f (α^{- 1} (y)) = α^{- 1} (y + 1) .

For a known function $f (x)$ , a problem is to solve the functional equation for the function $α -1 \equiv h$ , possibly satisfying additional requirements, such as $α -1 (0) = 1$ .

The change of variables $s α (x) = Ψ(x)$ , for a real parameter $s$ , brings Abel's equation into the celebrated Schröder's equation, $Ψ(f (x)) = s Ψ(x)$ .

The further change $F (x) = exp(s α (x))$ into Böttcher's equation, $F (f (x)) = F (x) s$ .

The Abel equation is a special case of (and easily generalizes to) the translation equation,

ω (ω (x, u), v) = ω (x, u + v),

e.g., for $ω (x, 1) = f (x)$ ,

ω (x, u) = α^{- 1} (α (x) + u)

. (Observe

ω (x,0) = x

.)

The Abel function $α (x)$ further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).

History

Initially, the equation in the more general form was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.

In the case of a linear transfer function, the solution is expressible compactly.

Special cases

The equation of tetration is a special case of Abel's equation, with $f = exp$ .

In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,

α (f (f (x))) = α (x) + 2,

and so on,

α (f_{n} (x)) = α (x) + n .

Solutions

The Abel equation has at least one solution on $E$ if and only if for all $x \in E$ and all $n \in ℕ^{*}$ , $f^{n} (x) \neq x$ , where $f^{n} = f \circ f \circ ... \circ f$ , is the function $f$ iterated $n$ times.

We have the following existence and uniqueness theorem^{: Theorem B}

Let $h : ℝ \to ℝ$ be analytic, meaning it has a Taylor expansion. To find: real analytic solutions $α : ℝ \to ℂ$ of the Abel equation $α \circ h = α + 1$ .

Existence

A real analytic solution $α$ exists if and only if both of the following conditions hold:

$h$ has no fixed points, meaning there is no $y \in ℝ$ such that $h (y) = y$ .
The set of critical points of $h$ , where $h^{'} (y) = 0$ , is bounded above if $h (y) > y$ for all $y$ , or bounded below if $h (y) < y$ for all $y$ .

Uniqueness

The solution is essentially unique in the sense that there exists a canonical solution $α_{0}$ with the following properties:

The set of critical points of $α_{0}$ is bounded above if $h (y) > y$ for all $y$ , or bounded below if $h (y) < y$ for all $y$ .
This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by

${α_{0} + β \circ α_{0} | β : ℝ \to ℝ is analytic, with period 1} .$

Approximate solution

Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point. The analytic solution is unique up to a constant.