AI 中文总结
本研究针对复平面上由两个相似映射生成的相似对动力系统，证明边界Mandelbrot集参数下吸引子边界的半群作用拓扑共轭于分段线性作用，给出填充集含割点的充要条件并推广了二次次要层状结构理论。
AI 中文摘要
“相似对”是复平面$\u2102$中由两个映射$f:z \to sz-1$和$g:z \to sz+1$（其中$|s|<1$）生成的动力系统，该系统对应一个吸引子$\Lambda$。Barnsley–Harrington Mandelbrot集$\mathcal{M}$是单位圆盘$\mathbb{D}$中使得$\Lambda$连通的所有$s$构成的集合。设$K$为连通吸引子$\Lambda$的填充集。对于边界$\partial \mathcal{M}$上的$s$，我们证明半群在$\partial K$上的（部分定义的）作用拓扑共轭于一类具有常斜率的（不连续）分段线性作用。在一个猜想（该猜想对$\partial \mathcal{M}$中“绝大多数”$s$成立）的条件下，我们给出了$K$包含割点的充要条件，该条件由$\partial K$上的动力学表述；同时我们通过显式递归算法得到有向图$\textrm{IG}$，用图中的无限行走描述了所有这类割点的集合。\n对于包含来自$\partial \mathcal{M}$中$s$对应作用的二维分段线性作用族，其“动力割点集”的结构恢复并推广了抽象Mandelbrot集的Douady–Hubbard–Thurston二次次要层状结构。
英文摘要
A {\em similarity pair} is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $Λ$. The Barnsley--Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $Λ$ is connected. Let $K$ denote the filled set of a connected $Λ$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady--Hubbard--Thurston quadratic minor lamination for the abstract Mandelbrot set.
Comments54 pages, 26 figures
