2025/04/23 by Gao, Yan, Wang, Xiaoguang, Wang, Yueyang · 2 citations
#37F15 #37F44 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 37F46 #Secondary 37F10
paper · doi:10.48550/arxiv.2504.16496
In this paper, we study the local connectivity and Hausdorff dimension for the boundaries of the bounded hyperbolic components in the space \mathcal Pd of polynomials of degree d≥ 3. It is shown that for any non disjoint-type bounded hyperbolic component \mathcal H⊂ \mathcal Pd, the locally connected part of ∂\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 ∂ \mathcal H: the local connectivity structure, the perturbation behavior, and the local Hausdorff dimensions.