Herman ring

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^[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<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, Michael Herman (1979^[2]) 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^[3]

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{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

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:^[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.

References

1 2 John Milnor, Dynamics in one complex variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.
↑ Herman, Michael-Robert (1979), "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations", Publications Mathématiques de l'IHÉS (49): 5–233, ISSN 1618-1913, MR 538680
↑ Omitted Values and Herman rings by Tarakanta Nayak.
↑ Mitsuhiro Shishikura, On the quasiconformal surgery of rational functions. Ann. Sci. Ecole Norm. Sup. (4) 20 (1987), no. 1, 1–29.
↑ Mitsuhiro Shishikura, Surgery of complex analytic dynamical systems, in "Dynamical Systems and Nonlinear Oscillations", Ed. by Giko Ikegami, World Scientific Advanced Series in Dynamical Systems, 1, World Scientific, 1986, 93–105.

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