Pseudoarcs, pseudocircles, Lakes of Wada and generic maps on \(S^2\)
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 singletons, pseudoarcs or pseudocircles. To show those results the authors also study a general technique which is effective to show that a nondegenerate hereditarily indecomposable \(P\)-like continuum \(M\) for some graph \(P\) is \(Q\)-like when \(Q\) is a certain type of a graph containing \(P\).
scientific article; zbMATH DE number 2173267