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Nonsingular star flows satisfy Axiom A and the no-cycle condition

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Abstract

We give an affirmative answer to a problem of Liao and Mañé which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C 1 neighborhood \(\mathcal{U}\) in the set of C 1 vector fields such that every singularity and every periodic orbit of every \(X\in\mathcal{U}\) is hyperbolic. We prove that any nonsingular star flow satisfies Axiom A and the no cycle condition.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Shaobo Gan & Lan Wen

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  1. Shaobo Gan
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  2. Lan Wen
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Correspondence to Shaobo Gan or Lan Wen.

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Dedicated to Shaotao Liao and Ricardo Mañé

Mathematics Subject Classification (2000)

37D30

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Gan, S., Wen, L. Nonsingular star flows satisfy Axiom A and the no-cycle condition. Invent. math. 164, 279–315 (2006). https://doi.org/10.1007/s00222-005-0479-3

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  • DOI: https://doi.org/10.1007/s00222-005-0479-3

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