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A non-flowable plane homeomorphism whose non-Hausdorff set consists of two disjoint lines. (English) Zbl 0857.54040
Summary: A homeomorphism $$f$$ of $$\mathbb{R}^2$$ is called flowable if there exists a topological flow whose time one map coincides with $$f$$. B. Kerékjártó [Ann. Scuola Norm. Sup. Pisa, II. Ser. 3, 393-400 (1934; Zbl 0010.03902)] constructed an orientation preserving homeomorphism of $$\mathbb{R}^2$$ without fixed points which is not flowable [M. Brown, Houston J. Math. 11, 455-469 (1985; Zbl 0605.57005)]. The reason why this homeomorphism is not flowable is that its non-Hausdorff set has branch points [the author, J. Math. Soc. Japan 47, 789-793 (1995; Zbl 0847.54011)]. In this paper, we construct an orientation preserving diffeomorphism of $$\mathbb{R}^2$$ without fixed points which is not flowable and whose non-Hausdorff set consists of two disjoint lines. This shows the difficulty in finding necessary and sufficient condition for flowability.

MSC:
 54H20 Topological dynamics (MSC2010) 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 57R50 Differential topological aspects of diffeomorphisms