论文标题
捕获双曲线组件的边界
Boundaries of capture hyperbolic components
论文作者
论文摘要
在复杂的动力学中，多项式图或有理图的全态家族中较高维度双曲线成分的边界是神秘的对象，其拓扑性和分析性能是基本问题。 在本文中，我们表明，在一些典型的多项式系列中（即由周期性批判关系定义的代数品种），捕获多余的成分$ \ mathcal h $的边界对球体是同质的，对球体是同型$ s^{2 \ dim_ \ dim_ \ dim_ \ dim_ \ nathbb {c}（c} c} c} c} c}（\ mathcalcal {此外，我们为$ \ partial \ Mathcal H $的Hausdorff维度建立了意外的身份：$$ \ pereratatorName {h {。} \ dim_ \ Mathbb {C}（\ Mathcal {h}） - 2+ \ max_ {f \ in \ partial \ Mathcal \ Mathcal {h}} \ operatorAtorName {h {。}。朱莉娅设定$ j（f）$的关键点。 在证明中，发现了一些具有独立兴趣的新结果。
In complex dynamics, the boundaries of higher dimensional hyperbolic components in holomorphic families of polynomials or rational maps are mysterious objects, whose topological and analytic properties are fundamental problems. In this paper, we show that in some typical families of polynomials (i.e. algebraic varieties defined by periodic critical relations), the boundary of a capture hyperbolic component $\mathcal H$ is homeomorphic to the sphere $S^{2\dim_\mathbb{C}(\mathcal{H})-1}$. Furthermore, we establish an unexpected identity for the Hausdorff dimension of $\partial \mathcal H$: $$\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.
