Empty Interior Recurrence for Continuous Flows on Surfaces

Abstract

In this paper we characterize topologically the empty interior subsets of a compact surface S which can be ω-limit sets of recurrent orbits (but not of nonrecurrent ones) of continuous flows on S. This culminates the classification of ω-limit sets for surface flows initiated in [Jiménez & Soler, 2001; Soler, 2003; Jiménez & Soler, 2004a, 2004b]. We also show that this type of ω-limit sets can always be realized (up to topological equivalence) by smooth flows but cannot be realized by analytic flows.

Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows

“…The reminder of the proof, that is W m is homeomorphic to a regular cylinder, is similar to that of the proof of Lemma 3.3 in [7]. Proof Let L be a non-proper orbit.…”

“…(2) Flows having infinitely many singularities and admitting an ω-limit set with empty interior and with infinitely many components in its complementary also exist. Moreover, this ω-limit set is not boundary of any regular cylinder, see [7].…”