Wild high-dimensional Cantor fences in ${\mathbb{R}}^n$, Part I

Abstract

Let $\mathcal{C}$ be the Cantor set. For each $n⩾3$ 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 $n⩾3$ 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 $\mathbb{A}(X\times \{s\})$, $s\in \mathcal{C}$, contains the corresponding $A(\mathcal{C}\times \{s\})$, and is ``nice'' on the complement $\mathbb{A}(X\times \{s\})-A(\mathcal{C}\times \{s\})$; 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).