Difference between revisions of "Abel equation"

From TORI
Jump to navigation Jump to search
(Follow advises by ChatGPT)
 
(One intermediate revision by the same user not shown)
Line 34: Line 34:
 
at least in some part of the complex plane, \(F=G^{-1}\) and \(G=F^{-1}\).
 
at least in some part of the complex plane, \(F=G^{-1}\) and \(G=F^{-1}\).
   
−
Together, the [[Abel function]] and the [[superfunction]] allow to express the \(c\)th [[iteration]] of the [[transfer function]] \(h\) as follows:
 +
Together, the [[Abel function]] and the [[superfunction]] allow to express the \(c\)th [[iteration]] of the [[transfer function]] \(T\) as follows:
 
: \((3)~ ~ ~ ~ ~ T^n(z)=F(n+G(z))\)
  
: \((3)~ ~ ~ ~ ~ T^n(z)=F(n+G(z))\)
 
which, at least for some values of \(z\), satisfies relation \(T^{n+m}(z) = T^n(T^m(z))\); in general, parameters \(n\) and \(m\) have no need to be integer. For the case of integer iterations, \(T^{-1}\) is inverse function of \(T~, ~ ~\)
  
which, at least for some values of \(z\), satisfies relation \(T^{n+m}(z) = T^n(T^m(z))\); in general, parameters \(n\) and \(m\) have no need to be integer. For the case of integer iterations, \(T^{-1}\) is inverse function of \(T~, ~ ~\)
 
\(T^0\) is identity function, \(T^1\!=\!T\) and so on.
  
\(T^0\) is identity function, \(T^1\!=\!T\) and so on.
 
It is assumed that both \(F\) and \(G\) are analytic in suitable domains, and the inversion is well-defined. In general, [[TORI]] do not deal with multivalued functions; so, the branch cut dividing the range of holomophism may destroy the harmony above. The branch cuts are unavoidable, as only in a trivial case both \(G\) and \(F=G^{-1}\) may be entire functions;
  
It is assumed that both \(F\) and \(G\) are analytic in suitable domains, and the inversion is well-defined. In general, [[TORI]] do not deal with multivalued functions; so, the branch cut dividing the range of holomophism may destroy the harmony above. The branch cuts are unavoidable, as only in a trivial case both \(G\) and \(F=G^{-1}\) may be entire functions;
−
Usually, at least one of functions \(F\),G\) has [[branch cut]]s.
 +
Usually, at least one of functions \(F\),\(G\) has [[branch cut]]s.
   
−
Once the [[Superfunction]] \(F\) and the corresponding [[Abel function]] \(G\) are specified, the transfer function \(T\) by (3) can be easily iterated arbitrary number of times, in particular, non-integer and even complex iteration is available.
 +
Once the [[Superfunction]] \(F\) and the corresponding [[Abel function]] \(G\) are specified in compatible domains, the transfer function \(T\) by (3) can be easily iterated arbitrary number of times, in particular, non-integer and even complex iterates are available.
   
 
To year 2026, Wikipedia <ref>
  
To year 2026, Wikipedia <ref>
Line 109: Line 109:
 
==Acknowledgment==
  
==Acknowledgment==
 
[[ChatGPT]] helped to improve this article.
  
[[ChatGPT]] helped to improve this article.
  +
==Notes by ChatGPT==
  +
Not all critics and suggestions by [[ChatGPT]] are taken into account in the version above. The "notes" are mentioned below.
  +
  +
The whole version of this article completely rewritten by [[ChatGPT]] is loaded as [[Abel equation.ChatGPT]].
  +
  +
===⚠ Over-strong claims===
  +
  +
1. **Existence of [[Abel function]]s for arbitrary growing real-holomorphic \(T\)**
  +
  +
You now correctly weakened the earlier universal claim, which is good.<br>
  +
However, the text still implicitly suggests broad solvability.
  +
  +
In the current state of iteration theory:
  +
* Existence is well understood **locally near fixed points** (e.g., via [[Kœnigs linearization]] for non-neutral [[fixed point]]s).
  +
* Global real-analytic solutions are delicate.
  +
* Entire solutions are exceptional.
  +
  +
It would be safer to explicitly say something like:
  +
  +
> Global existence of a real-holomorphic [[Abel function]] for an arbitrary growing real-holomorphic [[transfer function]] is not known in full generality; results are typically local or require additional hypotheses.
  +
  +
2. **“The [[Abel equation]] appears as a simplest equation”**
  +
  +
“Simplest” is philosophically true in your framework, but mathematically subjective.
  +
Better wording:
  +
  +
> The Abel equation corresponds to conjugation to the translation map \(z \mapsto z+1\), and can be viewed as a basic model of additive [[linearization]].
  +
  +
3. **Non-integer iteration always available**
  +
  +
You write:
  +
  +
> non-integer and even complex iteration is available.
  +
  +
This is true only **after** specifying domains, branches, and normalization.
  +
Without those, it sounds unconditional.
  +
  +
Suggestion:
  +
  +
> Once \(F\) and \(G\) are defined and analytic in compatible domains, fractional and complex iterates can be defined via formula (3).
  +
  +
===⚠ Terminological inconsistency===
  +
  +
You are consciously running two parallel terminologies:
  +
<pre>
  +
| Classical literature | TORI terminology |
  +
| ------------------------| ---------------- |
  +
| Abel equation | Abel equation |
  +
| Schröder equation | Schröder equation |
  +
| Böttcher equation | Böttcher equation |
  +
| solution of Abel eq. | Abel function |
  +
| solution of transfer eq.| Superfunction |
  +
</pre>
  +
That is acceptable — but you should clearly state once:
  +
  +
> In this article, the term *superfunction* denotes a solution of equation (2); this terminology is not standard in the broader literature.
  +
  +
Otherwise readers may believe this is universal terminology.
  +
  +
===⚠ Important mathematical clarification===
  +
  +
In classical iteration theory:
  +
<poem>
  +
[[Abel equation]] and [[Schröder equation]] are both *[[linearization problem]]s*.
  +
Abel corresponds to **neutral fixed point with multiplier 1**.
  +
Schröder corresponds to **multiplier \( \lambda \neq 1\)**.
  +
Böttcher corresponds to **superattracting fixed point**.
  +
</poem>
  +
Your article currently does not mention fixed points at all — but fixed points are the structural core of the theory.
  +
  +
Adding even 3–4 sentences about this would greatly increase mathematical credibility.
  +
  +
===⚠ Historical inaccuracy===
  +
  +
You write:
  +
  +
> He seems to mention the topic in 1881.
  +
  +
Two issues:
  +
  +
1. Abel died in 1829. 2. The 1881 reference is the posthumous edition of his collected works.
  +
  +
More importantly:
  +
  +
Abel did **not** formulate the modern Abel functional equation in its iteration-theoretic form. The name was attached later by the iteration community.
  +
  +
Safer formulation:
  +
  +
> The equation is named after Niels Henrik Abel; the terminology was introduced later in the development of iteration theory.
  +
  +
You already partially corrected this in the Warning section — good.
  +
===⚠ Conceptual gap===
  +
  +
The article still lacks:
  +
  +
Clear statement of **local vs global theory**
  +
  +
Clear statement of **normalization condition** (e.g. \(G(z_0)=0\))
  +
  +
Mention of **[[fixed point classification]]**
  +
  +
Adding even a short subsection:
  +
  +
> “Local theory near a fixed point”
  +
  +
would significantly improve the article.
   
 
==References==
  
==References==
Line 135: Line 241:
 
==Keywords==
  
==Keywords==
 
«[[Abel equation]]»,
  
«[[Abel equation]]»,
  +
«[[Abel equation.ChatGPT]]»,
 
«[[Abel function]]»,
  
«[[Abel function]]»,
 
«[[Abelfunction]]»,
  
«[[Abelfunction]]»,
 
«[[Asymptotic]]»,
  
«[[Asymptotic]]»,
  +
«[[ChatGPT]]»,
 
«[[Hellmuth Kneser]]»,
  
«[[Hellmuth Kneser]]»,
 
«[[Iterate]]»,
  
«[[Iterate]]»,
  +
«[[Linearization]]»,
 
«[[Niels Henrik Abel]]»,
  
«[[Niels Henrik Abel]]»,
 
«[[Regular iteration]]»,
  
«[[Regular iteration]]»,

Latest revision as of 23:26, 14 February 2026


Abel equation is functional equation that relates some known function (considered as transfer function) \(T\) to the corresponding Abel function \(G\) in the following way:

\((1)~ ~ ~ ~ ~ G(T(z))=G(z)+1\)

for \(z\) from some domain \(D\) in the complex plane. It is assumed that \(T\) is holomorphic (or real-analytic) function defined on a domain \(D\subset\mathbb C\). Often it is assumed also that along the real axis, \(T\) is growing function.

The solution \(G\) together with its inverse function \(F=G^{-1}\) allow to express the non-integer iterates of the transfer function \(T\). The examples of the transfer functions \(T\) and the solutions \(G\) are considered in book «Superfunctions» [1], 2020, although no general proof of existence of the real-holomorphic Abel functions \(G\) for arbitrary growing real-holomorphic transfer function \(T\) is provided.

The Abel equation appears as a simplest equation in the set or its conjugations; his set includes also the Schroeder equation, the Boettcher equation and other similar equations that have not yet established names.

The Abel equation is named after Niels Henrik Abel (see picture below). He seems to mention [2] the topic in 1881. It is difficult to verity: to year 2026, no free online version of paper [2] is found.

NielsHenrikAbel268x326.jpg
Niels Henrik Abel [3]

Transfer equation

The Abel equation is closely related to the transfer equation for the superfunction \(F\):

\( (2)~ ~ ~ ~ ~ F(z\!+\!1)=T(F(z))\)

The Abel function \(G\) is considered as inverse of the superfunction \(F\); at least in some part of the complex plane, \(F=G^{-1}\) and \(G=F^{-1}\).

Together, the Abel function and the superfunction allow to express the \(c\)th iteration of the transfer function \(T\) as follows:

\((3)~ ~ ~ ~ ~ T^n(z)=F(n+G(z))\)

which, at least for some values of \(z\), satisfies relation \(T^{n+m}(z) = T^n(T^m(z))\); in general, parameters \(n\) and \(m\) have no need to be integer. For the case of integer iterations, \(T^{-1}\) is inverse function of \(T~, ~ ~\) \(T^0\) is identity function, \(T^1\!=\!T\) and so on. It is assumed that both \(F\) and \(G\) are analytic in suitable domains, and the inversion is well-defined. In general, TORI do not deal with multivalued functions; so, the branch cut dividing the range of holomophism may destroy the harmony above. The branch cuts are unavoidable, as only in a trivial case both \(G\) and \(F=G^{-1}\) may be entire functions; Usually, at least one of functions \(F\),\(G\) has branch cuts.

Once the Superfunction \(F\) and the corresponding Abel function \(G\) are specified in compatible domains, the transfer function \(T\) by (3) can be easily iterated arbitrary number of times, in particular, non-integer and even complex iterates are available.

To year 2026, Wikipedia [4] makes no difference between the Abel equation (1) and the Transfer equation (2).

Sometimes it is useful, to have different names for the equation (1) and equation (2), and different names for their solutions. In TORI,
equation (1) is called Abel equation, and its solution is called Abelfunction, and
equation (2) is called Transfer equation (although term Transfer equation may have also other meaning(s)), and its solution is called Superfunction.

Uniqueness

For the transfer function \(T\) of general kind, the problem of existence and uniqueness of solution of the Abel equation is not trivial. Most of commonly used functions can be declared as transfer functions, and the corresponding Abel function can be constructed; better to say, many of them can be constructed. The additional conditions, for example, the asymptotic the infinity and the behavior in vicinity of the fixed points can be used to specify the unique solution [5][6].

Examples

The Abel equation becomes simple, if the transfer function \(T\) is considered as unknown, while the Transfer function \(F\) and its invese, id est the Abel function \(G\), are given. (in general any non-trivial function has many inverse functions). Then, the transfer function \(T\) can be expressed as follows:

\((4)~ ~ ~ ~ ~ T(z)=F(1+G(z))\)

Actually, such an expression is just a special case of equation (3) for \(n\!=\!1\).

The representation (4) allows to construct many examples. One can see that the division by a constant is Abel function of addition, logarithm is Abel function of addition and so on.

More examples are considered in article Transfer function and in book «Superfunctions»[1].

Conjugations

The Abel equation is one example of conjugating a function to a simpler dynamical model. The described conjugations list: translation, multiplication, power map, but not yet the exponential.

The Abel equation appears among various conjugations that include
the Abel equation \(\ G(T(z))=G(z)+1 \ \) ,
the Schroeder equation \(\ G(T(z))=s\, G(z) \ \) ,
the Boettcher equation \(\ G(T(z))= G(z)^k\)
and other similar equations that have not yet established names, for example,
the Tori equation \(\ G(T(z))= \exp_b(G(z))\ \), and, in general, for arbitrary function \(\Phi\)
the Phi equation \(\ G(T(z))= \Phi(G(z))\ \).

Warning

This article is uploaded at TORI in order to systematize the notations used in book «Superfunctions» [1].

Editor tries to follow the commonly used notations, but still, the interpretation suggested may deviate from those of other sites.

The equation is named after Niels Henrik Abel; the terminology originates from later developments in iteration theory.

Acknowledgment

ChatGPT helped to improve this article.

Notes by ChatGPT

Not all critics and suggestions by ChatGPT are taken into account in the version above. The "notes" are mentioned below.

The whole version of this article completely rewritten by ChatGPT is loaded as Abel equation.ChatGPT.

⚠ Over-strong claims

1. **Existence of Abel functions for arbitrary growing real-holomorphic \(T\)**

You now correctly weakened the earlier universal claim, which is good.
However, the text still implicitly suggests broad solvability.

In the current state of iteration theory:

  • Existence is well understood **locally near fixed points** (e.g., via Kœnigs linearization for non-neutral fixed points).
  • Global real-analytic solutions are delicate.
  • Entire solutions are exceptional.

It would be safer to explicitly say something like:

> Global existence of a real-holomorphic Abel function for an arbitrary growing real-holomorphic transfer function is not known in full generality; results are typically local or require additional hypotheses.

2. **“The Abel equation appears as a simplest equation”**

“Simplest” is philosophically true in your framework, but mathematically subjective. Better wording:

> The Abel equation corresponds to conjugation to the translation map \(z \mapsto z+1\), and can be viewed as a basic model of additive linearization.

3. **Non-integer iteration always available**

You write:

> non-integer and even complex iteration is available.

This is true only **after** specifying domains, branches, and normalization. Without those, it sounds unconditional.

Suggestion:

> Once \(F\) and \(G\) are defined and analytic in compatible domains, fractional and complex iterates can be defined via formula (3).

⚠ Terminological inconsistency

You are consciously running two parallel terminologies: 

| Classical literature    | TORI terminology  | 
| ------------------------| ----------------  | 
| Abel equation           | Abel equation     | 
| Schröder equation       | Schröder equation | 
| Böttcher equation       | Böttcher equation | 
| solution of Abel eq.    | Abel function     | 
| solution of transfer eq.| Superfunction     |

That is acceptable — but you should clearly state once:

> In this article, the term *superfunction* denotes a solution of equation (2); this terminology is not standard in the broader literature.

Otherwise readers may believe this is universal terminology.

⚠ Important mathematical clarification

In classical iteration theory:

Abel equation and Schröder equation are both *linearization problems*.
Abel corresponds to **neutral fixed point with multiplier 1**.
Schröder corresponds to **multiplier \( \lambda \neq 1\)**.
Böttcher corresponds to **superattracting fixed point**.

Your article currently does not mention fixed points at all — but fixed points are the structural core of the theory.

Adding even 3–4 sentences about this would greatly increase mathematical credibility.

⚠ Historical inaccuracy

You write:

> He seems to mention the topic in 1881.

Two issues:

1. Abel died in 1829. 2. The 1881 reference is the posthumous edition of his collected works.

More importantly:

Abel did **not** formulate the modern Abel functional equation in its iteration-theoretic form. The name was attached later by the iteration community.

Safer formulation:

> The equation is named after Niels Henrik Abel; the terminology was introduced later in the development of iteration theory.

You already partially corrected this in the Warning section — good.

⚠ Conceptual gap

The article still lacks:

Clear statement of **local vs global theory**

Clear statement of **normalization condition** (e.g. \(G(z_0)=0\))

Mention of **fixed point classification**

Adding even a short subsection:

> “Local theory near a fixed point”

would significantly improve the article.

References

  1. ↑ 1.0 1.1 1.2 https://nizugadro.mydns.jp/BOOK/486.pdf D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
  2. ↑ 2.0 2.1 Niels Henrik Abel. “Une équation d’un degré quelconque étant proposée, reconnaître si elle pourra être satisfaite algébriquement, ou non.”. Overs complètes, 1881, vol. 2, 330. (No free online version is found)
  3. ↑ https://commons.wikimedia.org/wiki/File:Niels_Henrik_Abel.jpg Description Niels Henrik Abel Source Originally uploaded to English wikipedia Painting by Johan Gørbitz (1782–1853)
  4. ↑ https://en.wikipedia.org/wiki/Abel_equation The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form \( f(h(x))=h(x+1)\) or \(\alpha (f(x))=\alpha (x)+1\). The forms are equivalent when α is invertible. h or α control the iteration of f.
  5. ↑ http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 H.Kneser. Reelle analytische Lösungen der Gleichung \(\varphi(\varphi(x))=e^x\). Equationes Mathematicae (Journal fur die reine und angewandte Mathematik) 187 56–67 (1950)
  6. ↑ http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, 81, p.65-76 (2011)

1998.oo.oo. http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf G.Belitskii, Yu.Lubich. The Abel equations and total solvability of linear functional equations. Studia Maghematica v.127 (1), 1998, p.81-97

2014.08.19. https://jbonet.webs.upv.es/wp-content/uploads/2016/05/Bonet_Domanski.pdf Jos ́e Bonet and Pawel􏰀 Doman ́ski. Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integr. Equ. Oper. Theory 81 (2015), 455–482 DOI 10.1007/s00020-014-2175-4. Published online August 19, 2014.

2015.04.30. http://jbonet.webs.upv.es/wp-content/uploads/2014/04/BD_eigenvaluessubmitted03032014.pdf Jose Bonet, Pawel Domanski. Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory, April 2015, Volume 81, Issue 4, pp 455–482.

Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=Abel_equation&oldid=46641"