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

José Ginés Espín Buendía; Víctor Jiménez Lopéz

doi:10.3934/dcdsb.2019010

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2019, Volume 24, Issue 3: 1143-1173. Doi: 10.3934/dcdsb.2019010

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

Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain

^* Corresponding author: V. Jiménez López
^* Corresponding author: V. Jiménez López

Received: September 2017

Revised: March 2018

Early access: January 2019

Published: January 2019

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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Abstract

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.

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

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Figure 1. The different parts of a shrub

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

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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)

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

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