论文标题
$ \ mathbb {r}^n $中的野生高维cantor围栏，第一部分
Wild high-dimensional Cantor fences in $\mathbb{R}^n$, Part I
论文作者
论文摘要
令$ \ Mathcal C $为Cantor集。 For each $n\geqslant 3$ we construct an embedding $A: \mathcal C \times \mathcal C \to \mathbb R^n$ such that $A(\mathcal C \times \{s\})$, for $s\in\mathcal C$, are pairwise ambiently incomparable everywhere wild Cantor sets (generalized Antoine's necklaces).这是本文证明的另一个新结果的基础：对于每个$ n \ geqslant 3 $和任何非空的完美紧凑型套件$ x $，可嵌入在$ \ mathbb r^{n-1} $中，我们描述了一个嵌入$ \ mathb a：x \ times \ mathcal c \ the $ mathb t y Mathb t y Mathb t y MathB \ the $ MathB r^n $ \ n $ \ Mathcal \ {s \}）$，$ s \ in \ Mathcal c $，包含相应的$ a（\ Mathcal C \ times \ {s \}）$，并且是``nice nice'''' \ {s \}）$;特别是，对于$ s \ in \ mathcal c $，图像$ \ mathbb a（x \ times \ {s \}）$是$ x $的$ s \ in \ mathcal c $。这概括并加强了J.R. Stallings（1960），R.B.Sher（1968）和B.L.Brechner-J.C.Mayer（1988）的定理。
Let $\mathcal C$ be the Cantor set. For each $n\geqslant 3$ we construct an embedding $A: \mathcal C \times \mathcal C \to \mathbb R^n$ such that $A(\mathcal C \times \{s\})$, for $s\in\mathcal C$, are pairwise ambiently incomparable everywhere wild Cantor sets (generalized Antoine's necklaces). This serves as a base for another new result proved in this paper: for each $n\geqslant 3$ and any non-empty perfect compact set $X$ which is embeddable in $\mathbb R^{n-1}$, we describe an embedding $\mathbb A : X \times \mathcal C \to \mathbb R^n$ such that each $\mathbb A (X \times \mathcal \{s\} )$, $s\in \mathcal C$, contains the corresponding $A (\mathcal C \times \{s\} )$, and is ``nice'' on the complement $\mathbb A (X \times \mathcal \{s\} )-A (\mathcal C \times \{s\} )$; in particular, the images $\mathbb A ( X \times \{s\})$, for $s\in\mathcal C$, are ambiently incomparable pairwise disjoint copies of $X$. This generalizes and strengthens theorems of J.R.Stallings (1960), R.B.Sher (1968), and B.L.Brechner-J.C.Mayer (1988).
