Uncountably many inequivalent Lipschitz homogeneous Cantor sets in ℝ³

Dennis Garity, Dušan Repovš and Matjaž Željko

Vol. 222 (2005), No. 2, 287–299

DOI: 10.2140/pjm.2005.222.287

Abstract

General techniques are developed for constructing Lipschitz homogeneous wild Cantor sets in ℝ³. These techniques, along with Kauffman’s version of the Jones polynomial and previous results on Antoine Cantor sets, are used to construct uncountably many topologically inequivalent such wild Cantor sets in ℝ³. This use of three-dimensional finite link invariants to detect distinctness among wild Cantor sets is unexpected. These Cantor sets have the same Antoine graphs and are Lipschitz homogeneous. As a corollary, there are uncountably many topologically inequivalent Cantor sets with the same Antoine graph.

Keywords

wild Cantor set, Lipschitz homogeneity, similitude, coefficient of similarity, defining sequence, link invariant

Mathematical Subject Classification 2000

Primary: 54E45, 54F65

Secondary: 57M30, 57N10

Milestones

Received: 13 April 2004

Accepted: 22 September 2004

Published: 1 December 2005