English: A rational function possesses Herman rings with period 2. Here the expression of this rational function is
where
This example was constructed by quasiconformal surgery[1]
from the quadratic polynomial
which possesses a Siegel disk with period 2. The parameters a, b, c are calculated by trial and error.
Letting
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.