File:Herman period=2.png

English: A rational function possesses Herman rings with period 2. Here the expression of this rational function is

${\displaystyle g_{a,b,c}(z)={\frac {z^{2}(z-a)}{z-b}}+c,\,}$

where

${\displaystyle {\begin{aligned}a&=0.17021425+0.12612303i,\\b&=0.17115266+0.12592514i,\\c&=1.18521775+0.16885254i.\end{aligned}}}$

This example was constructed by quasiconformal surgery^[1] from the quadratic polynomial

${\displaystyle h(z)=z^{2}-1-{\frac {e^{{\sqrt {5}}\pi i}}{4}}}$

which possesses a Siegel disk with period 2. The parameters a, b, c are calculated by trial and error.

Letting

${\displaystyle {\begin{aligned}a&=0.14285933+0.06404502i,\\b&=0.14362386+0.06461542i,{\text{ and}}\\c&=0.18242894+0.81957139i,\end{aligned}}}$

then the period of one of the Herman ring of g_a,b,c is 3.

Shishikura also given an example:^[2] a rational function which possesses a Herman ring with period 2, but the parameters showed above are different from his.

So there is a question: How to find the formulas of the rational functions which possess Herman rings with higher period?

According to the result of Shishikura, if a rational function ƒ possesses a Herman ring, then the degree of ƒ is at least 3. There also exist meromorphic functions that possess Herman rings.