# Difference between revisions of "Abel equation"

en>Michael Hardy |
en>Cuzkatzimhut m (→History) |
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:<math>\alpha(f(x))=\alpha(x)+1\!</math> | :<math>\alpha(f(x))=\alpha(x)+1\!</math> | ||

and | and controls the iteration of {{mvar|f}}. | ||

==Equivalence== | ==Equivalence== | ||

These equations are equivalent. Assuming that α is an [[invertible function]], the second equation can be written as | These equations are equivalent. Assuming that {{mvar|α}} is an [[invertible function]], the second equation can be written as | ||

:<math> \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\, .</math> | :<math> \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\, .</math> | ||

Taking | Taking {{math|''x'' {{=}} ''α''<sup>−1</sup>(''y'')}}, the equation can be written as | ||

::<math>f(\alpha^{-1}(y))=\alpha^{-1}(y+1)\, .</math> | ::<math>f(\alpha^{-1}(y))=\alpha^{-1}(y+1)\, .</math> | ||

For a function ''f''(''x'') assumed to be known, the task is to solve the functional equation for the function ''α''<sup>−1</sup>, possibly satisfying additional requirements, such as | For a function {{math|''f''(''x'')}} assumed to be known, the task is to solve the functional equation for the function {{math|''α''<sup>−1</sup>}}, possibly satisfying additional requirements, such as {{math|''α''<sup>−1</sup>(0) {{=}} 1}}. | ||

The change of variables ''s''<sup>''α''(''x'')</sup> = Ψ(''x''), for a real parameter | The change of variables {{math|''s''<sup>''α''(''x'')</sup> {{=}} Ψ(''x'')}}, for a real parameter {{mvar|s}}, brings Abel's equation into the celebrated [[Schröder's equation]], {{math|Ψ(''f''(''x'')) {{=}} ''s'' Ψ(''x'')}} . | ||

The further change ''F''(''x'') = exp(''s''<sup>''α''(''x'')</sup>) into [[Böttcher's equation]], ''F''(''f''(''x'')) = ''F''(''x'')<sup>''s''</sup>. | The further change {{math|''F''(''x'') {{=}} exp(''s''<sup>''α''(''x'')</sup>)}} into [[Böttcher's equation]], {{math|''F''(''f''(''x'')) {{=}} ''F''(''x'')<sup>''s''</sup>}}. | ||

==History== | ==History== | ||

Line 43: | Line 43: | ||

| coauthors= | | coauthors= | ||

| title=Theorems on functional equations | | title=Theorems on functional equations | ||

| journal= | | journal= Bull. Amer. Math. Soc. | ||

| volume= | | volume=19 | ||

| issue=2 | | issue=2 | ||

| pages= | | pages= 51-106 | ||

| year= | | year=1912 | ||

| doi=10.1090/S0002-9904-1912-02281-4 | |||

}}</ref> | }}</ref> | ||

was reported. | was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis. | ||

<ref name="U1">{{cite journal | <ref name="U1">{{cite journal | ||

| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf | | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf | ||

| author=G. Belitskii | | author=G. Belitskii|author2=Yu. Lubish | ||

| | |||

| title=The real-analytic solutions of the Abel functional equations | | title=The real-analytic solutions of the Abel functional equations | ||

| journal=[[Studia Mathematica]] | | journal=[[Studia Mathematica]] | ||

Line 68: | Line 68: | ||

| doi=10.1016/j.nahs.2006.04.002 | | doi=10.1016/j.nahs.2006.04.002 | ||

| author=Jitka Laitochová | | author=Jitka Laitochová | ||

| title =Group iteration for Abel’s functional equation | | title =Group iteration for Abel’s functional equation}} Studied is the Abel functional equation α(f(x))=α(x)+1</ref> | ||

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

In the case of linear transfer function, the solution can be expressed in compact form | |||

<ref name="linear">{{cite journal | <ref name="linear">{{cite journal | ||

| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf | | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf | ||

| author=G. Belitskii | | author=G. Belitskii|author2=Yu. Lubish | ||

| | | title=The Abel equation and total solvability of linear functional equations | ||

| title=The Abel equation and total solvability of linear functional | |||

| journal=[[Studia Mathematica]] | | journal=[[Studia Mathematica]] | ||

| volume=127 | | volume=127 | ||

Line 87: | Line 83: | ||

==Special cases== | ==Special cases== | ||

The equation of [[tetration]] is a special case of Abel's equation, with | The equation of [[tetration]] is a special case of Abel's equation, with {{math|''f'' {{=}} exp}}. | ||

In the case of an integer argument, the equation encodes a recurrent procedure, e.g., | |||

:<math>\alpha(f(f(x)))=\alpha(x)+2 ~,</math> | |||

and so on, | |||

:<math>\alpha(f_n(x))=\alpha(x)+n ~.</math> | |||

Fatou coordinates represent solutions of Abel's equation, describing local dynamics of discrete dynamical system near a [[Classification_of_Fatou_components|parabolic fixed point]].<ref>Dudko, Artem (2012). [http://www.math.toronto.edu/graduate/Dudko-thesis.pdf ''Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets''] Ph.D. Thesis </ref> | |||

==See also== | ==See also== | ||

*[[Functional equation]] | *[[Functional equation]] | ||

*[[ | *[[Iterated function]] | ||

*[[Schröder's equation]] | *[[Schröder's equation]] | ||

*[[Böttcher's equation]] | *[[Böttcher's equation]] | ||

*[[Infinite compositions of analytic functions]] | |||

==References== | ==References== | ||

<references/> | <references/> |

## Latest revision as of 15:14, 2 September 2014

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

or

and controls the iteration of Template:Mvar.

## Equivalence

These equations are equivalent. Assuming that Template:Mvar 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}, possibly satisfying additional requirements, such as *α*^{−1}(0) = 1.

The change of variables *s*^{α(x)} = Ψ(*x*), for a real parameter Template:Mvar, 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}.

## 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.,

and so on,

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

## See also

- Functional equation
- Iterated function
- Schröder's equation
- Böttcher's equation
- Infinite compositions of analytic functions

## References

- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }} Studied is the Abel functional equation α(f(x))=α(x)+1
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Dudko, Artem (2012).
*Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets*Ph.D. Thesis