A \(C^{\infty}\)-flow on \(S^ 3\) with a Denjoy minimal set
type: Item
instance of: scholarly article
title: A \(C^{\infty}\)-flow on \(S^ 3\) with a Denjoy minimal set
zbMATH Open document ID: 0539.58030
published in: Journal of Differential Geometry
publication date: 1981-01-01T00:00:00Z
review text: The main result of this paper is the construction of a \(C^{\infty}\) diffeomorphism of the annulus that leaves an embedded circle invariant and is such that the diffeomorphism restricted to the circle is a Denjoy map. A homeomorphism of the circle is called a Denjoy map if it has no periodic orbits and no dense orbits. In this example, the invariant circle is not isolated; there are periodic points arbitrarily near the circle. Recent papers by G. R. Hall, J. Harrison, and M. R. Herman all study annulus diffeomorphisms with Denjoy maps on invariant circles. Hall's example is \(C^{\infty}\) and the circle is an attractor. Hall's preprint refers to the other two authors. Harrison's example is \(C^ 2\) and the circle is semistable. (She also has written a preprint ''A \(C^ 2\)-counterexample to the Seifert conjecture''. The Seifert conjecture is what motivated the author of the paper under review.) Herman's example is \(C^{3-\epsilon}\) and is area preserving.
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 3858036
zbMATH Keywords: diffeomorphism of the annulus
zbMATH Keywords: Denjoy map
zbMATH Keywords: invariant circles
zbMATH Keywords: Seifert conjecture
Wikidata QID: Q115188400
author: Ronald J. Knill
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214436103
OpenAlex ID: W1584483920
DOI: 10.4310/JDG/1214436103
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description: scientific article; zbMATH DE number 3858036