Wild high-dimensional Cantor fences in R^n, Part Iстатья
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Аннотация:Let $С$ be the Cantor set. For each $n\ge 3$ we construct an embedding $A:C\times C\to R$ such that $A(C\times \{s\})$, for $s\in C$, 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 $n\ge 3$ and any non-empty perfect compact set $X$ which is embeddable in $R^{n-1}$, we describe an embedding $A:X\times C\to R^n$ such that each $A(X\times \{s\})$, $s\in C$, contains the corresponding $A(C\times \{s\})$ and is "nice" on the complement $A(X\times \{s\}) - $A(C\times \{s\})$; in particular, the images $A(X\times \{s\})$, for $s\in C$, 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).