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Espín Buendía, José Ginés; Jiménez Lopéz, Víctor

A topological characterization of the \(\omega\)-limit sets of analytic vector fields on open subsets of the sphere. (English) Zbl 1409.37052

Discrete Contin. Dyn. Syst., Ser. B 24, No. 3, 1143-1173 (2019).
Summary: In [Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 9, No. 2, 515–521 (2006; Zbl 1178.37015)], the second author and G. Soler López characterized, up to homeomorphism, the \(\omega\)-limit sets of analytic vector fields on the sphere and the projective plane. The authors also studied the same problem for open subsets of these surfaces.
Unfortunately, an essential lemma in their programme for general surfaces has a gap. Although the proof of this lemma can be amended in the case of the sphere, the plane, the projective plane and the projective plane minus one point (and therefore the characterizations for these surfaces in [the second author and G. Soler López, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 9, No. 2, 515–521 (2006; Zbl 1178.37015)] are correct), the lemma is not generally true, see [the authors, Qual. Theory Dyn. Syst. 16, No. 2, 293–298 (2017; Zbl 1385.37056)].
Consequently, the topological characterization for analytic vector fields on open subsets of the sphere and the projective plane is still pending. In this paper, we close this problem in the case of open subsets of the sphere.

Cited in 2 Documents

MSC:

37E35 Flows on surfaces
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C10 Dynamics induced by flows and semiflows

Keywords:

analytic vector field; \( \omega \)-limit set; topological classification

Citations:

Zbl 1178.37015; Zbl 1385.37056
PDFBibTeX XMLCite
\textit{J. G. Espín Buendía} and \textit{V. Jiménez Lopéz}, Discrete Contin. Dyn. Syst., Ser. B 24, No. 3, 1143--1173 (2019; Zbl 1409.37052)
Full Text: DOI arXiv

References:

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