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

TL;DR

This paper constructs complex wild Cantor sets in high-dimensional Euclidean spaces and extends classical embedding theorems to more general sets, revealing intricate topological structures.

Contribution

It introduces new embeddings of Cantor sets and arbitrary perfect sets in $ eal^n$, generalizing classical results and demonstrating the existence of pairwise incomparable wild Cantor sets.

Findings

Constructed embeddings of wild Cantor sets in $ eal^n$

Extended classical embedding theorems to broader classes of sets

Demonstrated pairwise incomparability of constructed sets

Abstract

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).