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.
Namely if ƒ possesses a Herman ring U with period p, then there exists a conformal mapping
and an irrational number , such that
So the dynamics on the Herman ring is simple.
It was introduced by, and later named after, Michael Herman (1979) 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
Here is an example of a rational function which possesses a Herman ring.
where such that the rotation number of ƒ on the unit circle is .
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.
Further, there is a rational function which possesses a Herman ring with period 2.
Here the expression of this rational function is
This example was constructed by quasiconformal surgery from the quadratic polynomial
then the period of one of the Herman ring of ga,b,c is 3.
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.
- 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.[full citation needed]
- 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.