Surgery on Herman rings of the standard Blaschke family

Let \begin{document} $B_{\alpha ,a}$ \end{document} be the Blaschke product of the following form: \begin{document}${B_{\alpha ,a}}(z) = {e^{2\pi {\rm{\mathbf{i}}}\alpha }}{z^{d + 1}}{(\frac{{z - a}}{{1 - az}})^d}.$ \end{document} If \begin{document} $B_{\alpha ,a}|_{S^1}$ \end{document} is analytically linearizable, then there is a Herman ring admitting the unit circle as an invariant curve in the dynamical plane of \begin{document} $B_{\alpha ,a}$ \end{document} . Given an irrational number \begin{document} $θ$ \end{document} , the parameters \begin{document} $(\alpha ,a)$ \end{document} such that \begin{document} $B_{\alpha ,a}|_{S^1}$ \end{document} has rotation number \begin{document} $θ$ \end{document} form a curve \begin{document} $T_d(θ)$ \end{document} in the parameter plane. Using quasiconformal surgery, we prove that if \begin{document} $θ$ \end{document} is of Brjuno type, the curve can be parameterized real analytically by the modulus of the Herman ring, from \begin{document} $a=M(θ)$ \end{document} up to \begin{document} $∞$ \end{document} with \begin{document} $M(θ)≥q 2d+1$ \end{document} , for which the Herman ring vanishes.Moreover, we can show that for a certain set of irrational numbers \begin{document} $θ ∈ \mathcal {B}\setminus\mathcal {H}$ \end{document} , the number \begin{document} $M(θ)$ \end{document} is strictly greater than \begin{document} $2d+1$ \end{document} and the boundary of the Herman rings consist of two quasicircles not containing any critical point.