Herman Rings of Elliptic Functions

It has been shown by Hawkins and Koss that over any given lattice, the Weierstrass $$\wp $$ function does not exhibit cycles of Herman rings. We show that, regardless of the lattice, any elliptic function of order two cannot have cycles of Herman rings. Through quasiconformal surgery, we obtain the existence of elliptic functions of order at least three with an invariant Herman ring. Finally, if an elliptic function has order $$o\ge 2$$ , then we show there can be at most $$o-2$$ invariant Herman rings.