# Abel equation

The Abel equation, named after Niels Henrik Abel, is a type of functional equation which can be written in the form

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

or, equivalently,

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

and controls the iteration of   f.

## Equivalence

These equations are equivalent. Assuming that α is an invertible function, the second equation can be written as

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

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

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

For a function f(x) assumed to be known, the task is to solve the functional equation for the function α−1h, possibly satisfying additional requirements, such as α−1(0) = 1.

The change of variables sα(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(sα(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).

## History

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.[4] [5][6]

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

## 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.,

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

and so on,

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

## Solutions

• formal solution : unique (to a constant)[8] (Not sure, because if ${\displaystyle u}$ is solution, then ${\displaystyle v(x)=u(x)+\Omega (u(x))}$, where ${\displaystyle \Omega (x+1)=\Omega (x)}$, is also solution[9].)
• analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point[10]
• Existence : Abel equation has at least one solution on ${\displaystyle E}$ if and only if ${\displaystyle \forall x\in E,\forall n\in \mathbb {N} ,f^{(n)}(x)\neq x}$, where ${\displaystyle f^{(n)}=f\circ f\circ ...\circ f}$, n times.[11]

Fatou coordinates describe local dynamics of discrete dynamical system near a parabolic fixed point.

## References

1. Aczél, János, (1966): Lectures on Functional Equations and Their Applications, Academic Press, reprinted by Dover Publications, ISBN 0486445232 .
2. Abel, N.H. (1826). "Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ..." Journal für die reine und angewandte Mathematik. 1: 11–15.
3. A. R. Schweitzer (1912). "Theorems on functional equations". Bull. Amer. Math. Soc. 19 (2): 51–106. doi:10.1090/S0002-9904-1912-02281-4.
4. Korkine, A (1882). "Sur un problème d'interpolation", Bull Sci Math & Astron 6(1) 228—242. online
5. G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations" (PDF). Studia Mathematica. 134 (2): 135–141.
6. Jitka Laitochová (2007). "Group iteration for Abel's functional equation". Nonlinear Analysis: Hybrid Systems. 1 (1): 95–102. doi:10.1016/j.nahs.2006.04.002.
7. G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations" (PDF). Studia Mathematica. 127: 81–89.
8. Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia
9. R. Tambs Lyche,ÉTUDES SUR L'ÉQUATION FONCTIONNELLE D'ABEL DANS LE CAS DES FONCTIONS RÉELLES., University of Trondlyim, Norvege
10. Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis
11. R. Tambs Lyche,Sur l'équation fonctionnelle d'Abel, University of Trondlyim, Norvege