2008
DOI: 10.1080/14689360802162872
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Abstract: We consider a compact 3-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C 1residual (dense G δ ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M . This is the flowsetting counterpart of Newhouse's Theorem 1.3 [17]. Our result follows from two theorems, the first one is the 3-dimensional continuous-time version of a theorem of Xia [21]… Show more

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Cited by 12 publications
(25 citation statements)
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“…We prove that there exists an open and dense subset of the incompressible 3-flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Mañé (see [26,13,9]) and of Newhouse (see [30,10]) for flows with singularities. That is we obtain for a residual subset of the C 1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.…”
mentioning
confidence: 82%
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“…We prove that there exists an open and dense subset of the incompressible 3-flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Mañé (see [26,13,9]) and of Newhouse (see [30,10]) for flows with singularities. That is we obtain for a residual subset of the C 1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.…”
mentioning
confidence: 82%
“…Here we complete the results of [9] and [10] fully extending the dichotomy from generic non-singular vector fields to generic vector fields in the family of C 1 all incompressible flows on 3-manifolds.…”
Section: Introductionmentioning
confidence: 99%
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“…The above theorem is proved in [9] (see [10] for divergence-free 3-flows) by looking first at the case of hyperbolic closed orbits with a small angle between the stable and unstable directions. Those are then showed to become elliptic by a small perturbation.…”
Section: Theorem 8 ([9]) Let D = 2 Given ε > 0 and An Open Subset Umentioning
confidence: 99%
“…Those are then showed to become elliptic by a small perturbation. On the other hand, for hyperbolic closed orbits with large angles and without dominated splitting, an adaptation of Mañé's perturbation techniques [10] leads again to elliptic orbits by a perturbation. The remaining case of hyperbolic closed orbits with dominated splitting and large angle is not true generically (as the case of parabolic ones).…”
Section: Theorem 8 ([9]) Let D = 2 Given ε > 0 and An Open Subset Umentioning
confidence: 99%