DCDS-B
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
Discrete & Continuous Dynamical Systems - B 2019, 24(3): 1143-1173 doi: 10.3934/dcdsb.2019010

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.

keywords: Analytic vector field $ \omega $-limit set open set sphere topological classification
DCDS
Existence of minimal flows on nonorientable surfaces
José Ginés Espín Buendía Daniel Peralta-salas Gabriel Soler López
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4191-4211 doi: 10.3934/dcds.2017178

Surfaces admitting flows all whose orbits are dense are called minimal. Minimal orientable surfaces were characterized by J.C. Benière in 1998, leaving open the nonorientable case. This paper fills this gap providing a characterization of minimal nonorientable surfaces of finite genus. We also construct an example of a minimal nonorientable surface with infinite genus and conjecture that any nonorientable surface without combinatorial boundary is minimal.

keywords: Flow surface minimality transitivity interval and circle exchange transformations suspension

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