Boundary of the central hyperbolic component II: boundary extension theorem

Abstract

In this paper, we study the boundary behavior of Milnor's parameterization $\Phi: \mathcal B_d\rightarrow \mathcal H_d$ 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 $D\in \partial \mathcal B_d$ which allows $\Phi$-extension. Further we show that cusps are dense in a full Hausdorff dimensional subset of $\partial \mathcal H_d$, partially confirming a conjecture of McMullen.