Separatrix conditions yielding either periodic orbits or unusal behavior for flows on \(M^ 3\)
type: Item
instance of: scholarly article
title: Separatrix conditions yielding either periodic orbits or unusal behavior for flows on \(M^ 3\)
zbMATH Open document ID: 0539.58029
DOI: 10.1007/BF02188030
author: D. R. Naugler
author: Carl S. Hartzman
published in: Aequationes Mathematicae
publication date: 1981-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/136908
review text: 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.
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 3858035
zbMATH Keywords: separatrix set
zbMATH Keywords: invariant solid torus
zbMATH Keywords: Seifert conjecture
zbMATH Keywords: higher order separatrices
zbMATH Keywords: minimal tori
MaRDI profile type: Publication
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OpenAlex ID: W4250994330
description: scientific article; zbMATH DE number 3858035