Separatrix conditions yielding either periodic orbits or unusal behavior for flows on \(M^ 3\)
This note provides sufficient conditions on the ''separatrix set'' of a \(C^ r\)-flow (on a compact manifold), for the existence of an invariant solid torus. The existence of such tori relates to the famous Seifert conjecture that all nonsingular \(C^ r\)-flows on \(S^ 3\) have closed orbits. For instance, F. W. Wilson, jun. pointed out [\textit{P. B. Percell} and \textit{F. W. Wilson jun.} Trans. Am. Math. Soc. 233, 93-103 (1977; Zbl 0367.34037)] that Schweitzer's Denjoy orbit-blocker can be ''plugged'' into an invariant solid torus, providing a \(C^ 1\)-flow with no periodic orbits. Two such solid tori may then be glued together to produce a \(C^ 1\)-counterexample to the Seifert conjecture. The definitions of the relevant terminology used by the authors are technical. But the ''separatrix sets'' are assumed to satisfy three hypotheses: each is topologically imbedded, there are no ''higher order separatrices'' and there are no n-dimensional regions of separatrices. The conditions on such separatrices, which guarantee the existence of invariant solid tori, are strong enough to guarantee the existence of periodic orbits in these tori, unless there are certain nonisolated minimal tori. With this motivation, neighborhoods of nonisolated invariant sets are also studied.
scientific article; zbMATH DE number 3858035