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Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows

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Abstract

We study \(C^1\)-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices. We prove that these flows exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets whose basins form a residual subset of the ambient manifold. This represents a partial positive answer to conjectures in Arbieto and Morales (Proc Am Math Soc 141:2817–2827, 2013), the Palis conjecture Palis (Nonlinearity 21:T37–T43, 2008) and gives a flow version of Crovisier and Pujals (Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, 2010).

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Notes

  1. Some related results have been appearing during the submission of this paper [30].

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 , Rio de Janeiro, Brazil

    A. Arbieto, C. A. Morales & B. Santiago

Authors
  1. A. Arbieto
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  2. C. A. Morales
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  3. B. Santiago
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Correspondence to C. A. Morales.

Additional information

Partially supported by CNPq, FAPERJ and PRONEX/DS from Brazil.

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Arbieto, A., Morales, C.A. & Santiago, B. Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows. Math. Ann. 361, 67–75 (2015). https://doi.org/10.1007/s00208-014-1061-3

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  • DOI: https://doi.org/10.1007/s00208-014-1061-3

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