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

TL;DR

This paper provides a topological characterization of omega-limit sets for analytic vector fields on open subsets of the sphere, resolving a previously identified gap in the theory.

Contribution

It completes the topological classification of omega-limit sets for analytic vector fields on open subsets of the sphere, addressing a gap in earlier work.

Findings

Provides a complete topological characterization for the sphere case.

Closes the gap in the previous proof for open subsets of the sphere.

Advances understanding of omega-limit sets in dynamical systems on surfaces.

Abstract

In [15], V. Jimenez 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 [8] are correct), the lemma is not generally true, see [15]. 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.