# 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. ^
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