A dichotomy for three-dimensional vector fields

Morales, C. A.; Pacífico, Maria José

doi:10.1017/s0143385702001621

Cited by 51 publications

(73 citation statements)

References 11 publications

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“…A dynamic property is called C 1 generic if it holds in a residual subset of X(M). The following is found in ( [29] (Thereom A)).…”

Section: Definitionmentioning

confidence: 82%

A Type of the Shadowing Properties for Generic View Points

Lee

2018

Axioms

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Abstract:We show that if a C 1 generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C 1 generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles.

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“…A dynamic property is called C 1 generic if it holds in a residual subset of X(M). The following is found in ( [29] (Thereom A)).…”

Section: Definitionmentioning

confidence: 82%

A Type of the Shadowing Properties for Generic View Points

Lee

2018

Axioms

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“…If the conjecture in ] mentioned above were true, this class would contain also the singular-Axiom A vector fields defined in [Morales and Pacifico 2003]. In any case there are many singularhyperbolic vector fields which are also Kupka-Smale.…”

Section: Figurementioning

confidence: 95%

“…An example of a singularhyperbolic vector field in S 3 which is not Kupka-Smale can be derived from the example described before. An example of a singular-hyperbolic vector field in S 3 satisfying the hypotheses of the next corollary can be found in [Morales and Pacifico 2003]. …”

Section: Figurementioning

confidence: 99%

A spectral decomposition for singular-hyperbolic sets

Morales¹,

Pacífico²

2007

Pacific J. Math.

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We extend the Spectral Decomposition Theorem for hyperbolic sets to singular-hyperbolic sets on 3-manifolds. We prove that an attracting singularhyperbolic set with dense periodic orbits and a unique equilibrium of a C r vector field, where r ≥ 1, is a finite union of transitive sets; the union is disjoint or the set contains finitely many distinct homoclinic classes. If the vector field is C r -generic, the union is in fact disjoint.

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“…Their result implies that a generic flow on a 3-manifold has an attractor or a repeller. This is done in [34], a paper that should be widely read. They raise the following question in Conjecture 1.3.…”

Section: Flowsmentioning

confidence: 99%

“…vector field exhibiting a homoclinic tangency or by a singular Axiom A one without cycles? (See [34] for definitions and details.) Their conjecture is "yes".…”

Section: Flowsmentioning

confidence: 99%

The topology and dynamics of flows

Sullivan¹

2007

Open Problems in Topology II

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Abstract. After a brief survey of various types of flows (Morse-Smale, Smale, Anosov, & partially hyperbolic) we focus on Smale flows on S 3 . However, we do give some consideration to Smale flows on other three-manifolds and to Smale diffeomorphisms. FlowsLet M be a compact connected Riemannian manifold without boundary. Let || · || be the norm on the tangent bundle T M and d(·, ·) the metric induced on M . By a flow on M we mean a smooth function f :Much of what we describe in this sections for flows carries over with suitable modifications to diffeomorphisms.The chain recurrent set of a flow f isThe chain recurrent set of a flow is said to have a hyperbolic structure if the tangent bundle of the manifold structure can be written as a Whitney sums of sub-bundles invariant under Df where E c x is the subspace of T M x corresponding to the orbit of x and such that there are constants C > 0 and λ > 0 for which ||Df t (v)|| ≤ Ce −λt ||x|| for v ∈ E s , t ≥ 0 and ||Df t (v)|| ≥ 1/Ce λt ||x|| for v ∈ E u , t ≥ 0. Steve Smale showed that when R is hyperbolic it is the closure of the periodic orbits of the flow. Smale also showed that when R is hyperbolic it has a finite decomposition into compact invariant sets called basic sets:Theorem (Spectral Decomposition Theorem). Suppose that the chain recurrent set R of a flow has a hyperbolic structure. Then R is a finite disjoint union of compact invariant sets Λ 1 , Λ 2 , . . . , Λ k where each Λ i contains an orbit that is dense in Λ i .We define respectively the stable and unstable manifolds of an orbit O in a flow f . W s (O) = {y ∈ M | d(f (y, t), f (x, t)) → 0 as t → ∞ for some x ∈ O} W u (O) = {y ∈ M | d(f (y, t), f (x, t)) → 0 as t → −∞ for some x ∈ O}

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A dichotomy for three-dimensional vector fields

Cited by 51 publications

References 11 publications

A Type of the Shadowing Properties for Generic View Points

A Type of the Shadowing Properties for Generic View Points

A spectral decomposition for singular-hyperbolic sets

The topology and dynamics of flows

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