Schroeder equation.ChatGPT

Schroeder equation is a functional equation of the form

\((1)~ ~ ~ ~ ~ G(T(z))=\lambda\, G(z)\)

where \(T\) is a given function (called here a transfer function), \(G\) is an unknown function, and \(\lambda\in\mathbb C\) is a constant.

The equation expresses the fact that \(T\) is conjugate to the multiplication map \(z\mapsto \lambda z\).

Relation to fixed points

The Schroeder equation arises naturally in the study of local dynamics near a fixed point \(z_0\) of \(T\): \[ T(z_0)=z_0 . \]

If \(T\) is analytic and \[ T'(z_0)=\lambda , \] with \(\lambda\neq 0\) and \(\lambda\neq 1\), then under suitable conditions there exists a function \(G\), analytic near \(z_0\), such that \[ G(z_0)=0, \quad G'(z_0)\neq 0, \] and equation (1) holds locally.

Thus the dynamics of \(T\) near \(z_0\) becomes conjugate to multiplication by \(\lambda\).

Linearization

Equation (1) is an example of linearization: it transforms a nonlinear map \(T\) into the linear map \(z\mapsto \lambda z\).

For comparison:

• The Abel equation corresponds to conjugation to the translation \(z\mapsto z+1\) and typically appears when \(T'(z_0)=1\).
• The Boettcher equation corresponds to conjugation to the power map \(z\mapsto z^k\) and appears near superattracting fixed points.

All three equations describe different types of local behavior near fixed points.

Iteration

If a solution \(G\) of (1) is invertible in some domain and \(F=G^{-1}\), then the iterates of \(T\) can be expressed as \[ T^n(z)=F(\lambda^n G(z)) \] for integer \(n\), and more generally \[ T^c(z)=F(\lambda^c G(z)) \] for complex \(c\), provided the branches of the power function are chosen consistently.

Thus the Schroeder equation provides a method for defining fractional and complex iterates of \(T\) near a fixed point.

Existence and uniqueness

Local analytic solutions of the Schroeder equation are well understood in many cases.

If \(0<|\lambda|<1\) (attracting fixed point), then the linearization exists locally and is unique up to multiplication by a nonzero constant.

If \(|\lambda|>1\) (repelling fixed point), similar results hold locally.

If \(|\lambda|=1\), the situation is more delicate and may depend on arithmetic properties of \(\lambda\).

Uniqueness typically requires a normalization condition such as fixing \(G'(z_0)=1\).

Global analytic solutions generally require additional assumptions.

Terminology

In classical literature, equation (1) is known as the Schroeder equation (also spelled Schröder equation). It is named after Ernst Schröder.

Historical remark

The equation was introduced in the study of functional iteration in the 19th century. It became a central tool in complex dynamics and the theory of iteration of analytic functions.

Warning

This article follows terminology compatible with that used in TORI. In particular, the term transfer function refers simply to the function being iterated.

Notes by Editor

The article above is generated by ChatGPT and saved "as is".

The References, Keywords and Categories below are added by Editor.

References

https://en.wikipedia.org/wiki/Schr%C3%B6der%27s_equation

https://en.wikipedia.org/wiki/Ernst_Schr%C3%B6der_(mathematician)

Keywords

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