圆作为拓扑分形
The circle as a topological fractal
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中文总结 AI 辅助
本文证明两个连续自映射不足以使圆成为拓扑分形，而三个映射即可，且对任意尺度存在两个映射覆盖圆，但无固定对适用于所有尺度。
中文摘要 AI 辅助
我们证明不存在由两个连续自映射组成的族能够见证圆是拓扑分形，从而回答了Karasová和本文作者提出的一个问题。已知三个映射足以实现这一点，因此该界限是最优的。相反，对于每个ε>0，存在依赖于ε的两个圆的连续自映射，其像覆盖整个圆，并且存在整数N，使得任意N个映射的复合像的直径小于ε。因此，两个映射在任何给定的尺度下都足够，但没有任何固定的映射对在所有尺度下都有效。
英文摘要
We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.
发表机构
- Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
机构由 AI 辅助整理，请以论文原文为准。
