A non-flowable plane homeomorphism whose non-Hausdorff set consists of two disjoint lines. (English) Zbl 0857.54040

Nakayama, Hiromichi

Houston J. Math. 21, No. 3, 569-572 (1995).

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

Keywords:

flowable homeomorphisms; orientation preserving diffeomorphism; flowability

Citations:

Zbl 0010.03902; Zbl 0605.57005; Zbl 0847.54011