Abel equation

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

or, equivalently,

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

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

For a function f(x) assumed to be known, the task 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]

e.g., for ,

.     (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.,

and so on,

Solutions

• formal solution : unique (to a constant)^[8]
• analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point^[9]

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

See also

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

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. ↑ Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis