Axes conventions

The Julia set of the cubic rational function e^2πitz²(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. 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

$\phi :U\rightarrow \{\zeta :0<r<|\zeta |<1\}$

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, Template:Harvs who first found and constructed this type of Fatou component.

Examples

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

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

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

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

This example was constructed by quasiconformal surgery 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 a, b, c are calculated by trial and error.

Letting

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

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.

References

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