Boundaries of the bounded hyperbolic components of polynomials

In this paper, we study the local connectivity and Hausdorff dimension for the boundaries of the bounded hyperbolic components in the space ${\mathcal{P}}_d$ of polynomials of degree $d\ge 3$. It is shown that for any non disjoint-type bounded hyperbolic component $\mathcal{H}\subset {\mathcal{P}}_d$, the locally connected part of $\mathrm{\partial }\mathcal{H}$, along each regular boundary strata, has full Hausdorff dimension $2d-2$. An essential innovation in our argument involves analyzing how the canonical parameterization of the hyperbolic component–realized via Blaschke products over a mapping scheme–extends to the boundary. This framework allows us to study three key aspects of $\mathrm{\partial }\mathcal{H}$: the local connectivity structure, the perturbation behavior, and the local Hausdorff dimensions.