Here is an example of a rational function which possesses a Herman ring

${\displaystyle f(z)={\frac {e^{2\pi i\tau }z^{2}(z-4)}{1-4z}}}$

where ${\displaystyle \tau =0.6151732\dots }$ such that the rotation number of ƒ on the unit circle is ${\displaystyle ({\sqrt {5}}-1)/2}$.

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.

Herman ring of period 1 and rotation number the golden mean.

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  image of the same Julia set, with src code but worse quality

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