Abel equation

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Short description: Equation for function that computes iterated values

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

[math]\displaystyle{ f(h(x)) = h(x + 1) }[/math]

or

[math]\displaystyle{ \alpha(f(x)) = \alpha(x)+1 }[/math].

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

Equivalence

The second equation can be written

[math]\displaystyle{ \alpha^{-1}(\alpha(f(x))) = \alpha^{-1}(\alpha(x)+1)\, . }[/math]

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

[math]\displaystyle{ f(\alpha^{-1}(y)) = \alpha^{-1}(y+1)\, . }[/math]

For a known function f(x) , a problem 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]

[math]\displaystyle{ \omega( \omega(x,u),v)=\omega(x,u+v) ~, }[/math]

e.g., for [math]\displaystyle{ \omega(x,1) = f(x) }[/math],

[math]\displaystyle{ \omega(x,u) = \alpha^{-1}(\alpha(x)+u) }[/math].     (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.,

[math]\displaystyle{ \alpha(f(f(x)))=\alpha(x)+2 ~, }[/math]

and so on,

[math]\displaystyle{ \alpha(f_n(x))=\alpha(x)+n ~. }[/math]

Solutions

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

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.[9] The analytic solution is unique up to a constant.[10]

See also

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. 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
  5. G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations". 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. 
  7. G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations". 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
  9. Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis
  10. Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia