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,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c99bfb88388bda1908e1fce83fd4d78e16f278)
where
![{\displaystyle {\begin{aligned}a&=0.17021425+0.12612303i,\\b&=0.17115266+0.12592514i,\\c&=1.18521775+0.16885254i.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a817c2f44add4e5fb38803bebe9c07a89c4f06e)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa20650788905ff481682fb254940564963962f3)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d93c1de700db53104b2983a86eb2e3245ece5d05)
then the period of one of the Herman ring of ga,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.