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Equivalence

The second equation can be written

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

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

f ( α − 1 ( y ) ) = α − 1 ( y + 1 ) . {\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 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

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

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

ω ( x , u ) = α − 1 ( α ( x ) + u ) {\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).

See also: Iterated function § Abelian property and Iteration sequences

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.⁴⁵⁶

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.,

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

and so on,

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

Solutions

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

We have the following existence and uniqueness theorem⁹: Theorem B 

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

Existence

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

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

Uniqueness

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

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

{ α 0 + β ∘ α 0 | β : R → R  is analytic, with period 1 } . {\displaystyle \{\alpha _{0}+\beta \circ \alpha _{0}|\beta :\mathbb {R} \to \mathbb {R} {\text{ 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.¹⁰ The analytic solution is unique up to a constant.¹¹

See also

• Functional equation
  • Schröder's equation
  • Böttcher's equation
• Infinite compositions of analytic functions
• Iterated function
• Shift operator
• Superfunction

• M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968).
• M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990.

References

1. Aczél, János, (1966): Lectures on Functional Equations and Their Applications, Academic Press, reprinted by Dover Publications, ISBN 0486445232 . /wiki/J%C3%A1nos_Acz%C3%A9l_(mathematician) ↩

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. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6 ↩

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. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183421988&view=body&content-type=pdf_1 ↩

4. Korkine, A (1882). "Sur un problème d'interpolation", Bull Sci Math & Astron 6(1) 228—242. online http://archive.numdam.org/ARCHIVE/BSMA/BSMA_1882_2_6_1/BSMA_1882_2_6_1_228_1/BSMA_1882_2_6_1_228_1.pdf ↩

5. G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations" (PDF). Studia Mathematica. 134 (2): 135–141. http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf ↩

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. /wiki/Doi_(identifier) ↩

7. G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations" (PDF). Studia Mathematica. 127: 81–89. http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf ↩

8. R. Tambs Lyche, Sur l'équation fonctionnelle d'Abel, University of Trondlyim, Norvege http://matwbn.icm.edu.pl/ksiazki/fm/fm5/fm5132.pdf ↩

9. Bonet, José; Domański, Paweł (April 2015). "Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions". Integral Equations and Operator Theory. 81 (4): 455–482. doi:10.1007/s00020-014-2175-4. hdl:10251/71248. ISSN 0378-620X. http://link.springer.com/10.1007/s00020-014-2175-4 ↩

10. Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis http://www.math.toronto.edu/graduate/Dudko-thesis.pdf ↩

11. Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia https://www.birs.ca/workshops/2015/15w5082/files/resman.pdf ↩