Pseudoarcs, pseudocircles, Lakes of Wada and generic maps on \(S^2\)
type: Item
instance of: scholarly article
title: Pseudoarcs, pseudocircles, Lakes of Wada and generic maps on \(S^2\)
zbMATH Open document ID: 1082.54014
author: Zoltán Buczolich
author: Udayan B. Darji
published in: Topology and its Applications
publication date: 2005-06-01T00:00:00Z
review text: The authors first prove a Bruckner-Grag type theorem for the fiber structure of generic maps from a compactum \(X\) into the unit interval \(\mathbb{I}\). Namely for a generic map \(f:X \to \mathbb{I}\) there is a countable dense subset \(D\) of \((\min f(X),\max f(X))\) such that if \(y \in D\), the space \(\text{Comp}(f^{-1}(y))\) of all components of \(f^{-1}(y)\) is homeomorphic to the union of the Cantor set and an isolated point, if \(y \in (\min f(X),\max f(X))\setminus D\), \(Comp(f^{-1}(y))\) is homeomorphic to the Cantor set, and if \(y = \min f(X)\) or \(\max f(X)\), \(f^{-1}(y)\) is a singleton. Here a generic map \(f \in C(X,\mathbb{I})\) is said to satisfy property \(P\) if the set \(\{f\in C(X,\mathbb{I}):f \) satisfies property \(P\}\) is residual in \(C(X,\mathbb{I})\). Hence the authors showed that the set \(\{f\in C(X,\mathbb{I}):f\) satisfy the above property\} contains a dense \(G_{\delta}\)-subset. \textit{J. Krasinkiewicz} [Bull. Pol. Acad. Sci., Math. 44, No. 2, 147--156 (1996; Zbl 0867.54020)] and \textit{M. Levin} [Fundam. Math. 151, No. 1, 47-5-2 (1996; Zbl 0860.54028)] showed that for a compact metric space \(X\) a generic map \(f:X \to \mathbb{I}\) has the property that all components of each of the fibers are hereditarily indecomposable. Here the authors investigate the special case of \(X = S^2\) and prove analogous results. Thus, a generic map \(f \in C(S^2,\mathbb{I})\) has the property that each component of each fiber of \(f\) is either a singleton or a figure-eight-like hereditarily indecomposable continuum. A surprising result is that a generic map \(f \in C(S^2,\mathbb{I})\) has the property that there is a countable dense subset \(D \subset f(S^2)\) such that for all \(y \in D\) there is a component of \(f^{-1}(y)\) homeomorphic to a Lakes of Wada continuum. Moreover a generic map \(f \in C(S^2,\mathbb{I})\) has the property that for almost all \(y \in (\min f(S^2),\max f(S^2))\) all components of \(f^{-1}(y)\) are either singl…
reviewed by: Akira Koyama
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 57N99
zbMATH DE Number: 2173267
zbMATH Keywords: generic maps
zbMATH Keywords: pseudoarcs
zbMATH Keywords: pseudocircles
zbMATH Keywords: Lakes of Wada
MaRDI profile type: Publication
full work available at URL: j.topol.2004.11.012
OpenAlex ID: W1998850434
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DOI: 10.1016/J.TOPOL.2004.11.012
description: scientific article; zbMATH DE number 2173267