Taming the Cantor fence
type: Item
instance of: scholarly article
title: Taming the Cantor fence
zbMATH Open document ID: 0933.54030
DOI: 10.1016/S0166-8641(97)00035-7
author: Russell B. Walker
author: E. D. Tymchatyn
published in: Topology and its Applications
publication date: 2000-04-04T00:00:00Z
review text: Let \(C\) be a compact and totally disconnected subset of the real line \(\mathbb{R}\). The authors prove that all topological embeddings \(C\times [0,1] \to\mathbb{R}^2\) are tame; that is there exists an ambient homeomorphism which straightens and makes parallel all arc components. More precisely, the main result of the paper is expressed by Theorem 1: If \(C\) is a Cantor set in \(\mathbb{R}\) and \(h:C\times [0,1]\to \mathbb{R}^2\) is a topological embedding then there exists an orientation preserving homeomorphism \(F:\mathbb{R}^2 \to\mathbb{R}^2\) such that \(F\circ h(C\times [0,1])= C\times [0,1]\). The proof is based on a lemma concerning sets separating the space between two sets (a set \(E\) separates \(\mathbb{R}^2\) between \(A\) and \(B\) if \(\mathbb{R}^2 \smallsetminus E\) is the disjoint union of two open sets \(U\) and \(V\) such that \(A\subset U\), \(B\subset V)\) which says: Let \(X\) be a compact subset of \(\mathbb{R}^2\), let \(A\) and \(B\) be disjoint continua in \(X\) and let \(M\) be an at most zero-dimensional compact subset of \(X \smallsetminus (A\cup B)\) such that \(X\) is separated between \(A\) and \(B\) by \(M\). Then there exists a simple closed curve \(\gamma\) such that \(\mathbb{R}^2 \smallsetminus \gamma\) is separated between \(A\) and \(B\) and \(\gamma\cap X\subset M\). The curve \(\gamma\) can be chosen so that \(A\) lies in the bounded complementary component of \(\gamma\) if \(B\) does not lie in a bound complementary component of \(A\). From the main result of the paper it follows that no positive entropy map of \(C\times [0,1]\) (which covers a homeomorphism of \(C)\) can be embedded into a near homeomorphism of \(\mathbb{R}^2\).
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 1308048
zbMATH Keywords: tame embedding
zbMATH Keywords: Cantor set
zbMATH Keywords: extension
zbMATH Keywords: sets separating the space between two sets
reviewed by: Andrzej Pelczar
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308048