Well I’ll pawn you my watch Pawn you my chain Pawn you my gold and diamond ring If this train roll me right I’ll be home tomorrow night ‘Cause I’m 900 miles from my home Lord I hate to hear that lonesome whistle blow
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.