Herman ring

In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou component where the rational function is conformally conjugate to an irrational rotation of the standard annulus.
Formal definition.
Namely if "ƒ" possesses a Herman ring "U" with period "p", then there exists a conformal mapping
and an irrational number formula_2, such that
So the dynamics on the Herman ring is simple.
Name.
It was introduced by, and later named after, Michael Herman (1979) who first found and constructed this type of Fatou component.
Examples.
Here is an example of a rational function which possesses a Herman ring.
where formula_5 such that the rotation number of "ƒ" on the
unit circle is formula_6.
The picture shown on the right is the Julia set of "ƒ": the curves in the white annulus are the orbits of some points under the iterations of "ƒ" while the dashed line denotes the unit circle.
There is an example of rational function that possesses a Herman ring, and some periodic parabolic Fatou components at the same time.
Further, there is a rational function which possesses a Herman ring with period 2.
Here the expression of this rational function is
where
This example was constructed by quasiconformal surgery
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 "g""a","b","c" is 3.
Shishikura also given an example: 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.
Herman rings for transcendental meromorphic functions have been studied by T. Nayak. According to a result of Nayak, if there is an omitted value for such a function then Herman rings of period 1 or 2 do not exist. Also, it is proved that if there is only a single pole and at least an omitted value, the function has no Herman ring of any period.