Abel equation.ChatGPT

Abel equation is a functional equation of the form

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

where \(T\) is a given function (called here a transfer function) and \(G\) is an unknown function (called here an Abel function). The equation is considered on some domain \(D\subset\mathbb C\) where \(T\) is defined and analytic (holomorphic or real-analytic).

The equation expresses the fact that the function \(T\) is conjugate to the translation \(z\mapsto z+1\).

Relation to iteration

If an Abel function \(G\) is invertible in some domain and \(F=G^{-1}\), then the function \(F\) satisfies the transfer equation

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

The functions \(F\) and \(G\) allow to define fractional and complex iterates of \(T\) by

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

provided all expressions are defined and analytic in compatible domains.

For integer \(n\), this construction coincides with the usual iteration: \(T^0\) is the identity, \(T^1=T\), \(T^{-1}\) is the inverse function (if it exists), and \(T^{n+m}=T^n\!\circ T^m\).

Local theory near fixed points

In classical iteration theory, the Abel equation appears in the study of local dynamics near a fixed point \(z_0\) of \(T\), that is, \(T(z_0)=z_0\).

If \(T'(z_0)=1\) (neutral fixed point with multiplier 1), then under suitable conditions the map can be conjugated locally to a translation, leading to the Abel equation.

For comparison:

• The Schroeder equation corresponds to a fixed point with multiplier \( \lambda \neq 1 \).
• The Boettcher equation corresponds to a superattracting fixed point.

All three equations are examples of linearization problems.

Existence and uniqueness

Existence and uniqueness of solutions of the Abel equation depend strongly on the properties of the transfer function \(T\).

Local analytic solutions near fixed points are well understood in many cases.

Global real-analytic or entire solutions are much more delicate and require additional hypotheses. In general, global existence of a real-holomorphic Abel function for an arbitrary growing real-holomorphic transfer function is not known in full generality.

Uniqueness typically requires normalization conditions, for example fixing the value of \(G\) at a chosen point.

Relevant results include work of Hellmuth Kneser and later developments in iteration theory.

Terminology

In this article, the term Superfunction is used for a solution of equation (2). This terminology is not standard in the broader literature and is specific to the conventions adopted at TORI.

Historical remark

The equation is named after Niels Henrik Abel. The modern functional equation used in iteration theory was formulated later; the terminology originates from subsequent developments in complex dynamics and functional equations.

Warning

This article follows the terminology system used in book «Superfunctions». The notation may differ from that used in other sources.

Notes by Editor

The article above is loaded as an alternative of article «Abel equation».

This version is expected to be used in future to compile some hybrid version. It is difficult to arrange while many related articles are not yet uploaded; even some drafts are not yet prepared.

The keywords below are added by Editor in order to simplify the search and the referencing.

References

Keywords

«Abel equation», «Abel function», «Abelfunction», «Asymptotic», «ChatGPT», «Hellmuth Kneser», «Iterate», «Niels Henrik Abel», «Regular iteration», «Superfunction», «Superfunctions», «Transfer equation», «Transfer function»,