Abel equation

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This article is about certain functional equations. For ordinary differential equations that are cubic in the unknown function, see Abel equation of the first kind.

The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the form

$f(h(x)) = h(x + 1)\,\!$

or

$\alpha(f(x))=\alpha(x)+1\!$

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

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

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

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

For a function f(x) assumed to be known, the task is to solve the functional equation for the function α−1, 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.

History

Initially, the equation in the more general form [1] [2] was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis. [3][4]

In the case of a linear transfer function, the solution can be expressed in compact form. [5]

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.,

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

and so on,

$\alpha(f_n(x))=\alpha(x)+n ~.$

Fatou coordinates represent solutions of Abel's equation, describing local dynamics of discrete dynamical system near a parabolic fixed point.[6]

References

1. ^
2. ^ A. R. Schweitzer (1912). "Theorems on functional equations". Bull. Amer. Math. Soc. 19 (2): 51–106. doi:10.1090/S0002-9904-1912-02281-4.
3. ^ G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations". Studia Mathematica 134 (2): 135–141.
4. ^ 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. Studied is the Abel functional equation α(f(x))=α(x)+1
5. ^ G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations". Studia Mathematica 127: 81–89.
6. ^ Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis