Boundaries of the bounded hyperbolic components of polynomials

Abstract: 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 $d\geq 3$. It is shown that for any non disjoint-type bounded hyperbolic component $\mathcal H\subset \mathcal P_d$, the locally connected part of $\partial\mathcal H$, 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 $\partial \mathcal H$: the local connectivity structure, the perturbation behavior, and the local Hausdorff dimensions.