Some remarks on the \(\omega\)-limit sets for plane, sphere and projective plane analytic flows. (English) Zbl 1385.37056
Summary: We show that Lemma 4.6 (and hence Proposition 4.9) in [the third author and J. Llibre, Adv. Math. 216, No. 2, 677–710 (2007; Zbl 1134.37016)] has an essential gap. We amend this gap in the particular cases of the sphere, the plane, the projective plane and the projective plane minus one point, thus the main results in [loc. cit.] remain correct. On the other hand, some counterexamples for the sphere minus two points and for the projective plane minus two points are given; as a consequence, Theorems 7.1 and 7.2 in [loc. cit.] are incomplete.
MSC:
| 37E35 | Flows on surfaces |
| 37C10 | Dynamics induced by flows and semiflows |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
Citations:
Zbl 1134.37016References:
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| [2] | Jiménez López, V., Llibre, J.: A topological characterization of the \[\omega\] ω-limit sets for analytic flows on the plane, the sphere and the projective plane. Adv. Math. 216, 677-710 (2007) · Zbl 1134.37016 |
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