Herman ring

In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou component[1] where the rational function is conformally conjugate to an irrational rotation of the standard annulus.

Namely if ƒ possesses a Herman ring U with period p, then there exists a conformal mapping and an irrational number

It was introduced by, and later named after, Michael Herman (1979[2]) who first found and constructed this type of Fatou component.

There is an example of rational function that possesses a Herman ring, and some periodic parabolic Fatou components at the same time.

Here the expression of this rational function is where This example was constructed by quasiconformal surgery[4] from the quadratic polynomial which possesses a Siegel disk with period 2.

Shishikura also given an example:[5] 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?

These denominators can be identified by the sequence of nodes along the edge of the Mandelbrot set approaching the point.

Similarly, Herman rings can be identified in a Mandelbrot set of rational functions by observing a series of nodes arranged on both sides of a curve, and choosing points along that curve, avoiding the attached nodes, thereby obtaining a desired sequence of denominators in the continued fraction expansion of the rotation number.