File:Ernst schroederFragment.jpg

Ernst Schroeder.

Fragment from image http://mizugadro.mydns.jp/t/index.php/File:Ernst_schroeder.jpg

Portrait of the German logician and mathematician Ernst Schroeder (Ernst Schröder). The photo was taken between 1890 and 1902. ^[1]

Original filename: https://upload.wikimedia.org/wikipedia/commons/3/34/Ernst_schroeder.jpg

This image is used as Fig.6.3 at page 67 of book «Superfunctions» ^[2][3]
in order to attribute the Schroeder equation:
In Regular iteration, the Schroeder equation appears as analogy of the Transfer equation; the Schroeder function appears as an analogy of the Superfunction.
For trunsfer function \(T\) with fixed point zero (id est, \(T(0)\!=\!0\)), the pair (Schroederfunction \(S\), ArcSchroeder \(S^{-1}\)) allows to express iterates of \(T\) in the similar way, as it can be done with pair (Superfunction, Abelfunction):

\[ T^n(z)=S(n\ (S^{-1}(z)) \]

In this expression, number \(n\) of iterate has no need to be integer.

The similar expression through the Superfunction \(F\) and the Abelfunction \(G\!=\!F^{-1}\) is

\[ T^n(z)=F(n+G(z)) \]

References

Keywords

«Ernst Schroeder», «Iterate», «Regular iteration», «Schroeder equation», «Schroederfunction», «Superfunctions», «Transfer function», «Transferfunction»,

«Суперфункции», «Уравнение Шредера»,