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Surgery on Herman rings of the standard Blaschke family

The author is supported by NSFC (grant No. 11426177,11301417) and NSF of Northwest University (grant No. NC14035)
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  • Let $B_{\alpha ,a}$ be the Blaschke product of the following form:

    ${B_{\alpha ,a}}(z) = {e^{2\pi {\rm{\mathbf{i}}}\alpha }}{z^{d + 1}}{(\frac{{z - a}}{{1 - az}})^d}.$

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

    Mathematics Subject Classification: Primary:37F50;Secondary:37F10.


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