Where t=.6141866…; a = ¼; b = 0, 0405353 + 0, 0255082i ; such that the rotation number of f(t,b,a) on the unit circle is (√5−1)/2 . The image has been rotated.
The Julia set of the cubic rational function e2πitz2(z−4)/(1−4z) with t=.6151732… chosen so that the rotation number is (√5−1)/2, which has a Herman ring (shaded).
If f is a rational function, defined in the extended complex plane, and if it is a nonlinear function (degree > 1) then for a periodic component of the Fatou set, exactly one of the following holds:
U contains an attracting periodic point.
U is parabolic.
U is a Siegel disc.
U is a Herman ring: 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.