The rational maps $z\mapsto 1+1/\omega z^d$ have no Herman rings
The rational maps $z\mapsto 1+1/\omega z^d$ have no Herman rings
We prove that for every $d \in \mathbb {N}, d \geq 2$, the rational maps in the family $\{ z \mapsto 1 + 1/\omega z^d : \omega \in \textbf {C} \setminus \{0 \} \}$ have no Herman rings. From this we conclude a dynamical characterization for the parameters in the …