Difference between revisions of "Abel equation"

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


and shows non-trivial properties at the iteration.
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  <math>~x=\alpha^{-1}(y)</math>, the equation can be written as
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&nbsp;α<sup>−1</sup>(0)&nbsp;=&nbsp;1.
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)&nbsp;{{=}}&nbsp;1}}.


The change of variables ''s''<sup>''α''(''x'')</sup> =  Ψ(''x''), for a real parameter ''s'', brings Abel's equation into the celebrated [[Schröder's equation]], Ψ(''f''(''x'')) =&nbsp;''s''&nbsp;Ψ(''x'') .
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''&nbsp;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==
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| coauthors=
| coauthors=
| title=Theorems on functional equations
| title=Theorems on functional equations
| journal=[[Bulletin des Sciences Mathématiques]]
| journal= Bull. Amer. Math. Soc.
| volume=27
| volume=19
| issue=2
| issue=2
| pages=31
| pages= 51-106
| year=1903
| year=1912
| doi=10.1090/S0002-9904-1912-02281-4
}}</ref>
}}</ref>
was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis
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
| coauthors=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>
| abstract=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
| coauthor=Yu. Lubish
| title=The Abel equation and total solvability of linear functional equations
| title=The Abel equation and total solvability of linear functional equtions
| 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 <math>f=\exp</math>. In the case of an integer argument, the equation is just a recurrent procedure.
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]]
*[[Abel function]]
*[[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

Template:Dablink

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 sα(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(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.,

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

References

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