Koenigs function

Koenigs function (Функция Кёнигса ) is normalized solution of the Schroeder equation.

Koenigs function is suggested as a tool for non-integer iterate of any holomorphic transfer function \(T\) such that \(T(0)=0\), \(s=T'(0)\ne 0\), \(s \ne 1\).

The Schroeder equation for the Koenigs function can be written as follows: \[ \mathrm{Koenigs}(T(z))=s\, \mathrm{Koenigs}(z) \]

Gabriel Xavier Paul Koenigs

Generalization

Sometimes, the generalization is used, displacing the fixed point of the Transfer function to some distance from zero, defying the new Transfer function

\[ \tau(z) = T(z-\mathrm{shift})+\mathrm{shift} \]

for some constant \(\mathrm{shift}\).

The Normalization may imply: \( ~ ~ ~ ~ \begin{array}{c} T(z_0)=z_0 \\ s=T'(z_0)\ne 0,1\\ K(z_0)=0\\ K'(z_0)=1 \end{array}\)

The resulting equation and its solution also can be qualified as Schroeder equation and Koenigs function.

This generalization may have sense for a transfer function that has various fixed points, and the regular iterates constructed at different fixed points have no need to coincide.

However, even in this case, for implementation of the iterates, it may have sense move the fixed point to zero, in order to deal each time with the simplest Koenigs function with fixed point zero, as it is assumed in the Preamble.

Notations

Term Koenigs function seems to be coined keeping in mind the publication , 1884, by Gabriel Xavier Paul Koenigs.

Various articles use different letters to denote the Transfer function \(T\) and the Koenigs function - even if the articles appear at the same site, at some Wikipedia.

This is unavoidable, as the number of functions to be considered greatly exceeds the bunker of letters of the Latin and Greek alphabets, and use of Hiragana and Kanji characters is not yet supported as some versions of Latex. In order to keep the deduction portable, it is strongly recommended to denote any function with either a single Italic letter or with a single Greek letter of use the explicit name in Roman font - at one does this with special functions (sin, cos, exp, erf, Factorial, etc.)

In TORI, by default, the Transfer function is denoted as \(T\).

The generic Koenigs function is dented with name Koenigs or, especially during the deduction, with any letter that is not loaded with another meaning.

As soon as some special function with some special name «なまえ» is considered, the specific name of the Koenig function is expected can be chosen as «Koなまえ» - in analogy as it is done with specific Superfunctions and Abelfunctions in book «Superfunctions».

In many cases, the inverse function of the Koenigs exist. To year 2026, yet, there is no established name for the Koenigs function; an there is need to generate it. ShatGPT dislikes notation "ArcKoenigs". Editor respects ShatGPT as a bold critic; so, perhaps, the function can be called «Schroeder» or «Schr», or any «Spunk», any «Кукарямба», that might randomly appear at the editor's heavy keyboard - at least until any more usual name for the function is detected.

Inverse function

Let \(f=\mathrm{Koenigs}^{-1}\)

Then \[\mathrm{Koenigs}(T(f(p)) =t\, p \] \[ T(f(p))=f(t p) \] This is scaling equation, analogy of the transfer equation for superfunctions. In the input of function \(f\) in the right hand side the input of the function is scaled with factor \(t\), instead of to be displaced, transferred for unity, as it takes place for the transfer equation.

The inverse of the Koenigs function conjugates \(T\) to multiplication.

Relations to other functions

The Koenigs function is closely related to several other functional equations arising in the theory of iteration.

Relation to the Schroeder equation

The Koenigs function is the normalized analytic solution of the Schroeder equation \[ G(T(z)) = s\, G(z), \] constructed near a fixed point \(z_0\) of the transfer function \(T\) with multiplier \(s=T'(z_0)\ne 0,1\).

Thus, the Koenigs function provides the local linearization of \(T\), conjugating it to the multiplication map \[ z \mapsto s z. \]

If \(K\) is the Koenigs function and \(f=K^{-1}\), then \[ T(f(p)) = f(sp), \] so the inverse Koenigs function conjugates \(T\) to scaling.

Relation to the Abel equation

The Abel equation \[ G(T(z)) = G(z)+1 \] can be regarded as a limiting case of the Schroeder equation when the multiplier \(s\to 1\).

Near a fixed point with \(T'(z_0)=1\) (parabolic case), linearization by multiplication is no longer possible, and conjugation to the translation map \(z\mapsto z+1\) is used instead.

Thus:

• Koenigs function — conjugation to multiplication.
• Abel function — conjugation to translation.

Relation to the Boettcher equation

If \(z_0\) is a superattracting fixed point of \(T\), that is, \[ T'(z_0)=0, \] then the appropriate conjugation is given by the Boettcher equation \[ G(T(z)) = G(z)^k, \] which conjugates \(T\) to a power map.

Hence the three classical equations correspond to three types of fixed points:

• Multiplier \(s\ne 0,1\) → Schroeder equation → Koenigs function.
• Multiplier \(s=1\) → Abel equation → Abel function.
• Multiplier \(s=0\) → Boettcher equation → Boettcher function.

Relation to fractional iteration

If the Koenigs function is invertible in some domain, then \[ T^c(z)=K^{-1}(s^c K(z)) \] defines fractional and complex iterates of \(T\), provided a consistent branch of the power function is chosen.

In this sense, the Koenigs function is a fundamental tool for constructing regular iteration near non-parabolic fixed points.

Conceptual position

Within the hierarchy of conjugations used in iteration theory, the Koenigs function occupies the position corresponding to linear semigroups of the form \[ p \mapsto s^c p. \]

Together with the Abel function (translation semigroup) and the Boettcher function (power semigroup), it forms one of the three classical mechanisms for reducing nonlinear iteration to simpler algebraic dynamics.

Appendices

Warning

The analysis suggested is prepared for a project of generalization of term "Asymptotic", and the notations are adjusted to this goal.

Editor and ChatGPT try to make the notations similar to those observed in internet, but not always this is achieved.
So the notations used in TORI may differ from the most common ones.

Acknowlegement

ChatGPT contributed in this article (section "Relation to other functions" and wordings).

Humor

Hints to remember the names of mathematicians who worked on the related topics:
Their ancestors used to böettch,
Their ancestors used to kneze,
Their ancestors used to schröd,
Their ancestors used to szek,
But ancestors of Neils were also Able to add unity to value of a function of to value of its input.
As for ancestors of Gabriel Xavier Paul Koenigs, they seem to establish the city Koenigsberg.
By the analogy, one can guess, that somewhere exist also come Vodker function (or even a Vödker function ), invented by a researcher whose ancestors used to vödka (it is difficult to believe that such a complicated topic could be developed without vodking), but Editor has not yet found publications confirming the hypothesis.
All the hypothesis above are based exclusively on the names; so, they need verifications before any serious use.

References

https://en.wikipedia.org/wiki/Koenigs_function .. Koenigs (1884) proved that there is a unique holomorphic function h defined on D, called the Koenigs function, such that h(0) = 0, h '(0) = 1 and Schröder's equation is satisfied, \( h(f(z))=f^{\prime}(0)h(z)~\)/

https://ru.wikipedia.org/wiki/Функция_Кёнигса Фу́нкция Кёнигса связана с решением функционального уравнения \(F[f(x)]=cF(x)\), где \(F(x)\) — неизвестная функция, \(f(x)\) и \(c\) — данные функция и константа. Обычно это уравнение (без особых исторических оснований) называют уравнением Шрёдера. ..

https://en.wikipedia.org/wiki/Schröder%27s_equation Schröder's equation,[1][2][3] named after Ernst Schröder, is a functional equation with one independent variable: given the function h, find the function Ψ such that \(\Psi \big(h(x)\big)=s\Psi (x)\) ..

https://zh.wikipedia.org/wiki/柯尼格斯函数 .. Koenigs (1884)证明了，在D上可以定义证明唯一的全纯函数h，称为柯尼希斯函数，使得 \(h(0)=0\), \(h'(0)=1\)，同时满足施罗德方程： .. \(h(f(z))=f^{\prime }(0)h(z)~\). ..

1958.oo.oo. https://projecteuclid.org/journals/acta-mathematica/volume-100/issue-3-4/Regular-iteration-of-real-and-complex-functions/10.1007/BF02559539.full Regular iteration of real and complex functions G. Szekeres Acta Math. 100(3-4): 203-258 (1958). DOI: 10.1007/BF02559539

2004.oo.oo. https://www.acadsci.fi/mathematica/Vol29/contrera.pdf Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 471–488 FIXED POINTS AND BOUNDARY BEHAVIOUR OF THE KOENIGS FUNCTION Manuel D. Contreras, Santiago D ́ıaz-Madrigal, and Christian Pommerenke .. σ ◦ φ = φ′(τ)σ // with σ′(τ) = 1, for every holomorphic self-map φ of D with inner DW-point and φ′(τ) ̸= 0. This map σ is called the Koenigs function associated to φ and σ is univalent, whenever φ is. \(\varphi)_n\big(\gamma(r)\big)=\sigma^{-1}\big(\gamma(r)\big)\) ..

Keywords

«Abel equation», «Abel equation.ChatGPT», «Abel function», «Abelfunction», «Asymptotic», «Boettcher equation», «ChatGPT», «Friedrich Wilhelm Karl Ernst Schröder», «Gabriel Xavier Paul Koenigs», «Hellmuth Kneser», «Inverse Koenigs function», «Iterate», «Koenigs function», «Linearization», «Niels Henrik Abel», «Regular iteration», «Schroeder equation», «Schroeder equation.ChatGPT», «Schroeder function», «Superfunction», «Superfunctions», «Transfer equation», «Transfer function»,