Boundaries of capture hyperbolic components

2022/06/15 by Jie Cao, Xiaoguang Wang, Cao, Jie +3
Mathematics · Physics and Astronomy · #37F10 #37F15 #37F35 (Secondary) #37F46 (Primary) #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2206.07462

openalex publication_date 2022/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^{2dim_ℂ(H)-1}. Furthermore, we establish an unexpected identity for the Hausdorff dimension of ∂ \mathcal H: \operatornameH.dim(\partialH) = 2 dim_ℂ(H)-2+max_{f∈\partialH} \operatornameH.dim(∂ 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.

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