Wild high-dimensional Cantor fences in [formula omitted], Part I

Abstract

Let C be the Cantor set. For each n⩾3 we construct an embedding A:C×C→Rn such that A(C×{s}), for s∈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 Rn−1, we describe an embedding A:X×C→Rn such that each A(X×{s}), s∈C, contains the corresponding A(C×{s}), and is “nice” on the complement A(X×{s})−A(C×{s}); in particular, the images A(X×{s}), for s∈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).