由复树生成的连通自相似集族
Families of connected self-similar sets generated by complex trees
摘要：
本文引入复树理论作为研究一类自相似集的新方法。用复树的尖对尖等价关系编码的方程组得到单参数连通自相似集族$F_\mathcal{A}(z)$。为了研究$F_\mathcal{A}(z)$在定义了这些族的区域$\mathcal{R}\subset\mathbb{C}$中的拓扑变化，我们引入了一种新的集合$\mathcal{M}\subseteq\mathcal{R}$，它扩展了通常的参数空间的连通性轨迹的概念.此外，我们考虑另一个集$\mathcal{M}_0\subseteq\mathcal{M}$有关的一种特殊类型的连接，我们提供了一个定理。除其他事项外，本理论提供了一个统一的框架，家庭的自相似集传统上研究作为单独的元素$F_\mathcal{A}(z)$断开参数$z\in\mathcal{R}\backslash\mathcal{M}$。
英文摘要：
The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected self-similar sets $F_\mathcal{A}(z)$. In order to study topological changes of $F_\mathcal{A}(z)$ in regions $\mathcal{R}\subset\mathbb{C}$ where these families are defined, we introduce a new kind of set $\mathcal{M}\subseteq\mathcal{R}$ which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set $\mathcal{M}_0\subseteq\mathcal{M}$ related to a special type of connectivity for which we provide a theorem. Among other things, the present theory provides a unified framework to families of self-similar sets traditionally studied as separate with elements $F_\mathcal{A}(z)$ disconnected for parameters $z\in\mathcal{R}\backslash\mathcal{M}$.
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