Abel equation

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

${\displaystyle f(h(x))=h(x+1)}$

or

${\displaystyle \alpha (f(x))=\alpha (x)+1}$.

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

The second equation can be written

${\displaystyle \alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.}$

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

${\displaystyle f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.}$

For a known function f(x) , a problem is to solve the functional equation for the function α⁻¹ ≡ 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,¹

${\displaystyle \omega (\omega (x,u),v)=\omega (x,u+v)~,}$

e.g., for ${\displaystyle \omega (x,1)=f(x)}$,

${\displaystyle \omega (x,u)=\alpha ^{-1}(\alpha (x)+u)}$.     (Observe ω(x,0) = x.)

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

Initially, the equation in the more general form ^{2 3} was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.⁴⁵⁶

In the case of a linear transfer function, the solution is expressible compactly.⁷

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.,

${\displaystyle \alpha (f(f(x)))=\alpha (x)+2~,}$

and so on,

${\displaystyle \alpha (f_{n}(x))=\alpha (x)+n~.}$

The Abel equation has at least one solution on ${\displaystyle E}$ if and only if for all ${\displaystyle x\in E}$ and all ${\displaystyle n\in \mathbb {N} ^{*}}$, ${\displaystyle f^{n}(x)\neq x}$, where ${\displaystyle f^{n}=f\circ f\circ ...\circ f}$, is the function f iterated n times.⁸

We have the following existence and uniqueness theorem^{9: Theorem B}

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

A real analytic solution ${\displaystyle \alpha }$ exists if and only if both of the following conditions hold:

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

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

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

$${\displaystyle \{\alpha _{0}+\beta \circ \alpha _{0}|\beta :\mathbb {R} \to \mathbb {R} {\text{ is analytic, with period 1}}\}.}$$

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.¹⁰ The analytic solution is unique up to a constant.¹¹