Axes conventions: Difference between revisions

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[[Image:Herman Standard.png|200px|right|thumb|The Julia set of the cubic rational function ''e''<sup>2&pi;''it''</sup>''z''<sup>2</sup>(''z''&minus;4)/(1&minus;4''z'') with ''t''=.6151732... chosen so that the rotation number is (&radic;5&minus;1)/2, which has a Herman ring (shaded).]]
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In the mathematical discipline known as [[complex dynamics]], the '''Herman ring''' is a [[Classification of Fatou components|Fatou component]].<ref name=dyn>[[John W. Milnor|John Milnor]], [http://books.google.com/books?id=DsthOelUMlkC&lpg=PR7&pg=PA161#v=onepage&f=false ''Dynamics in one complex variable'']: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.</ref> where the [[rational function]] is conformally conjugate to an irrational rotation of the standard [[annulus (mathematics)|annulus]].
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==Formal definition==
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Namely if ''&fnof;'' possesses a Herman ring ''U'' with period ''p'', then there exists a [[conformal mapping]]


:<math>\phi:U\rightarrow\{\zeta:0<r<|\zeta|<1\}</math>
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and an [[irrational number]] <math>\theta</math>, such that
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:<math>\phi\circ f^{\circ p}\circ\phi^{-1}(\zeta)=e^{2\pi i\theta}\zeta.</math>
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So the dynamics on the Herman ring is simple.
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It was introduced by, and later named after, {{harvs|txt|authorlink=Michael Herman (mathematician)|first=Michael|last= Herman|year=1979}} who first found and constructed this type of Fatou component.
 
==Examples==
Here is an example of a rational function which possesses a Herman ring.<ref name=dyn/>
 
:<math>f(z) = \frac{e^{2 \pi i \tau} z^2(z - 4)}{1 - 4z}</math>
 
where <math>\tau=0.6151732\dots</math> such that the [[rotation number]] of ''&fnof;'' on the
unit circle is <math>(\sqrt{5}-1)/2</math>.
 
The picture shown on the right is the [[Julia set]] of ''&fnof;'': the curves in the white annulus are the orbits of some points under the iterations of ''&fnof;'' while the dashed line denotes the unit circle.
 
There is an example of rational function that possesses a Herman ring, and some [[Classification of Fatou components|periodic parabolic Fatou components]] at the same time.
[[Image:Herman+Parabolic.png|600px|center|thumb|A rational function <math>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}</math> that possesses a Herman ring and some periodic parabolic Fatou components, where <math>t=0.6141866\dots,\,a=1/4,\,b=0.0405353-0.0255082i</math> such that the rotation number of <math>f_{t,a,b}</math> on the unit circle is <math>(\sqrt{5}-1)/2</math>. The image has been rotated.]]
 
Further, there is a rational function which possesses a Herman ring with period 2.
[[Image:Herman period=2.png|600px|center|thumb|A rational function possesses Herman rings with period 2]]
Here the expression of this rational function is
 
:<math> g_{a,b,c}(z) = \frac{z^2(z-a)}{z-b} + c, \, </math>
 
where
 
: <math>
\begin{align}
a & = 0.17021425+0.12612303i, \\
b & = 0.17115266+0.12592514i, \\
c & = 1.18521775+0.16885254i.
\end{align}
</math>
 
This example was constructed by quasiconformal surgery<ref>[[Mitsuhiro Shishikura]], [http://archive.numdam.org/article/ASENS_1987_4_20_1_1_0.pdf ''On the quasiconformal surgery of rational functions'']. Ann. Sci. Ecole Norm. Sup. (4) '''20''' (1987), no. 1, 1–29.</ref>
from the quadratic polynomial
 
:<math>h(z)=z^2 - 1 - \frac{e^{\sqrt{5}\pi i}}{4} </math>
 
which possesses a [[Siegel disk]] with period&nbsp;2. The parameters ''a'',&nbsp;''b'',&nbsp;''c'' are calculated by [[trial and error]].
 
Letting
 
: <math>
\begin{align}
a & = 0.14285933+0.06404502i, \\
b & = 0.14362386+0.06461542i,\text{ and} \\
c & = 0.18242894+0.81957139i,
\end{align}
</math>
 
then the period of one of the Herman ring of ''g''<sub>''a'',''b'',''c''</sub> is&nbsp;3.
 
[[Mitsuhiro Shishikura|Shishikura]] also given an example:<ref>[[Mitsuhiro Shishikura]], [http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/99208/1/0574-13.pdf ''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.</ref> 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 ''&fnof;'' possesses a Herman ring, then the degree of ''&fnof;'' is at least&nbsp;3. There also exist [[meromorphic function]]s that possess Herman rings.
 
==References==
{{reflist}}
*{{Citation | last1=Herman | first1=Michael-Robert | title=Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations | url=http://www.numdam.org/item?id=PMIHES_1979__49__5_0 | mr=538680 | year=1979 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=49 | pages=5–233}}
 
[[Category:Fractals]]
[[Category:Limit sets]]
[[Category:Complex dynamics]]

Latest revision as of 15:11, 8 February 2014

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