Boundary of the central hyperbolic component II: boundary extension theorem

In this paper, we study the boundary behavior of Milnor’s parameterization $\mathrm{\Phi }:{\mathcal{B}}_d\to {\mathcal{H}}_d$ of the central hyperbolic component ${\mathcal{H}}_d$ via Blaschke products. We establish a boundary extension theorem by giving a necessary and sufficient condition for $D\in \mathrm{\partial }{\mathcal{B}}_d$ which allows $\mathrm{\Phi }$-extension. Further we show that cusps are dense in a full Hausdorff dimensional subset of $\mathrm{\partial }{\mathcal{H}}_d$, partially confirming a conjecture of McMullen.