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.

Figures (32)

• Figure 1: The model plane and the dynamical plane
• Figure 2: Graphs in model plane and dynamical plane, for $f$
• Figure 3: Graphs in model plane and dynamical plane, for $g$. Note that $\Omega_{v,q}(D_g)$ is continuous in $g\in \mathcal{F}$, but $\phi_{g,v}(\Omega_{v,q}(D_g))$ is not continuous in $g\in \mathcal{F}$ in general.
• Figure 4: Some near degenerate critical points in model plane.
• Figure 5: Limbs and angles $\theta_{f,v}^{\pm}$

Theorems & Definitions (176)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Local connectivity
• Theorem 1.6: Local Hausdorff dimension
• Theorem 1.7: Perturbation on $\partial \mathcal{H}$
• Theorem 1.8
• Proposition 2.1
• Remark 2.2