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A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere

  • * Corresponding author: V. Jiménez López

    * Corresponding author: V. Jiménez López

This work has been partially supported by Ministerio de Economía y Competitividad, Spain, grant MTM2014-52920-P. The first author has been also supported by Fundación Séneca by means of the program "Contratos Predoctorales de Formación del Personal Investigador", grant 18910/FPI/13

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  • In [15], V. Jiménez López and J. Llibre 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 [15] are correct), the lemma is not generally true, see [6].

    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.

    Mathematics Subject Classification: Primary: 37E35, 37B99; Secondary: 37C10.


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  • Figure 1.  The different parts of a shrub

    Figure 2.  The $ 5 $-cusped hypocycloid (left) and the $ 8 $-cusped hypocycloid with some arcs (right)

    Figure 3.  In the positive (counterclockwise) sense: a node with a flexible (F) leaf, a rigid (R) sprig, a bland (B) leaf and a rigid leaf (left), and a leaf with flexible (f), bland (b), rigid (r) and bland nodes (right)

    Figure 4.  From left to right, the shrubs $ A $, $ A' $, $ A'' $ and $ A^* $.

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