Taming the Cantor fence
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 \(\mathbb{R}^2\).
sets separating the space between two sets
scientific article; zbMATH DE number 1308048