发表机构
School of Mathematics, Shanghai University of Finance and Economics（上海财经大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对二次族的非逃逸轨迹$\text{KM}$，证明其边界具有满豪斯多夫维数4，在双曲参数处局部连通，在部分实抛物参数处局部连通性不成立，还构造混合谜题片分析其局部连通性。
AI 中文摘要
针对二次族$f_c(z)=z^2+c$，我们研究非逃逸轨迹$\text{KM}=\bigl\{(c,z):c\in\boldsymbol{\text{M}},\thinspace z\in K_c\bigr\}$，它是Inou与Kiwi框架下三次捕获拉直的模型空间。其中$\boldsymbol{\text{M}}$为曼德博集合，$K_c$是$f_c$的填充朱利亚集，$J_c=\boldsymbol{\text{∂}}K_c$。我们证明$\boldsymbol{\text{∂}}\text{KM}$具有满豪斯多夫维数4；在双曲参数$c\in\boldsymbol{\text{M}}$处，$\text{KM}$在所有$(c,z)$点局部连通；在$\boldsymbol{\text{∂}}\boldsymbol{\text{M}}$上的参数处，我们从Yoccoz谜题和准谜题构造混合谜题片；对于非退化对，两个平面谜题的收缩意味着$\text{KM}$在$(c,z)$处局部连通；但在部分$c$为实抛物且$z\in K_c\boldsymbol{\text{\textbackslash}}J_c$的$(c,z)$点，即使参数空间和填充朱利亚纤维在该处均局部连通，$\text{KM}$的局部连通性仍不成立。
英文摘要
For the quadratic family $f_c(z)=z^2+c$, we study the non-escaping locus $\KM=\{(c,z):c\in\M,\ z\in K_c\}$, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here $\M$ is the Mandelbrot set, $K_c$ is the filled Julia set of $f_c$, and $J_c=\partial K_c$. We prove that $\partial\KM$ has full Hausdorff dimension $4$. The set $\KM$ is locally connected at every $(c,z)$ with hyperbolic $c\in\M$. At parameters on $\partial\M$, we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of $\KM$ at $(c,z)$. However, local connectedness fails at some $(c,z)$ with $c$ real parabolic and $z\in K_c\setminus J_c$, even though both the parameter space and the filled Julia fiber are locally connected there.
