Herman ring

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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[edit]

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[edit]

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

Function[edit]

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

Examples[edit]

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[edit]

  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.