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| In a recent paper by V. Jim\'{e}nez L\{'o}pez and J. Llibre (Adv. Math., 2007), topological characterizations of the $\omega$-limit sets for analytic flows on open subsets of the sphere and the projective plane were given. An auxiliary lemma, essential in the proof, states that analytic flows on arbitrary analytic surfaces have the following property: if an orbit meets both sides of an arc of singular points contained in its $\omega$-limit set, then the flow must be equally oriented in both sides.
We will show that, despite the fact that the property is true for the plane, the sphere and the projective plane, the statement is not true in general. We shall present a couple of examples on open proper subsets of the plane and on the torus. Therefore, the characterizations given in the cited paper are incomplete; in this talk we present full, correct characterizations of these sets.
(This is part of a joint work, still in progress, with V. Jim\'enez L\'opez). |
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