# Herman ring

The Julia set of the cubic rational function eitz2(z−4)/(1−4z) with t=.6151732... chosen so that the rotation number is (√5−1)/2, which has a Herman ring (shaded).

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.

## Formal definition

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

$\phi:U\rightarrow\{\zeta:0

and an irrational number $\theta$, such that

$\phi\circ f^{\circ p}\circ\phi^{-1}(\zeta)=e^{2\pi i\theta}\zeta.$

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.

## Function

• Polynomials do not have Herman rings.
• Rational functions can have Herman rings
• Transcendental entire maps do not have them[2]

## Examples

Here is an example of a rational function which possesses a Herman ring.[1]

$f(z) = \frac{e^{2 \pi i \tau} z^2(z - 4)}{1 - 4z}$

where $\tau=0.6151732\dots$ such that the rotation number of ƒ on the unit circle is $(\sqrt{5}-1)/2$.

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.

A rational function $f_{t,a,b}(z)=e^{2\pi it}z^3\,\frac{1-\overline{a}z}{z-a}\,\frac{1-\overline{b}z}{z-b}$ that possesses a Herman ring and some periodic parabolic Fatou components, where $t=0.6141866\dots,\,a=1/4,\,b=0.0405353-0.0255082i$ such that the rotation number of $f_{t,a,b}$ on the unit circle is $(\sqrt{5}-1)/2$. The image has been rotated.

Further, there is a rational function which possesses a Herman ring with period 2.

A rational function possesses Herman rings with period 2

Here the expression of this rational function is

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

where

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

This example was constructed by quasiconformal surgery[3] from the quadratic polynomial

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

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

Letting

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

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

Shishikura also given an example:[4] 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.

## References

1. ^ a b John Milnor, Dynamics in one complex variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.
2. ^ Omitted Values and Herman rings by Tarakanta Nayak
3. ^ Mitsuhiro Shishikura, On the quasiconformal surgery of rational functions. Ann. Sci. Ecole Norm. Sup. (4) 20 (1987), no. 1, 1–29.
4. ^ Mitsuhiro Shishikura, Surgery of complex analytic dynamical systems, in "Dynamical Systems and Nonlinear Oscillations", Ed. by Giko Ikegami, World Scientic Advanced Series in Dynamical Systems, 1, World Scientic, 1986, 93–105.