Herman ring

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

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

and an irrational number ${\displaystyle \theta }$, such that

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

So the dynamics on the Herman ring is simple.

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

• Polynomials do not have Herman rings.
• Rational functions can have Herman rings
• Transcendental entire maps do not have them[3]

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

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

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

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

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

where

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

This example was constructed by quasiconformal surgery[4] from the quadratic polynomial

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

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

Letting

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

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

Shishikura also given an example:[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?

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.

• Douady rabbit
• Siegel disc