Boundaries of capture hyperbolic components

TL;DR

This paper investigates the topological and dimensional properties of boundaries of capture hyperbolic components in polynomial families, revealing they are homeomorphic to spheres and establishing a formula for their Hausdorff dimension.

Contribution

It proves that these boundaries are topologically spheres and derives a novel formula relating their Hausdorff dimension to the dimensions of Fatou components.

Findings

Boundaries are homeomorphic to spheres of specific dimensions.

Derived a formula for Hausdorff dimension of boundaries.

Discovered new results with independent mathematical interest.

Abstract

In complex dynamics, the boundaries of higher dimensional hyperbolic components in holomorphic families of polynomials or rational maps are mysterious objects, whose topological and analytic properties are fundamental problems. In this paper, we show that in some typical families of polynomials (i.e. algebraic varieties defined by periodic critical relations), the boundary of a capture hyperbolic component $\mathcal H$ is homeomorphic to the sphere $S^{2\dim_\mathbb{C}(\mathcal{H})-1}$. Furthermore, we establish an unexpected identity for the Hausdorff dimension of $\partial \mathcal H$: $$\operatorname{H{.}dim}(\partial\mathcal{H}) = 2 \dim_\mathbb{C}(\mathcal{H})-2+\max_{f\in\partial\mathcal{H}} \operatorname{H{.}dim}(\partial A^J(f)),$$ where $A^J(f)$ is the union of the bounded attracting Fatou components of $f$ associated with the free critical points in the Julia set $J(f)$. In the proof, some new results with independent interests are discovered.