Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers

Morales, C. A.; Pacífico, Maria José; Pujals, Enrique R.

doi:10.4007/annals.2004.160.375

Cited by 148 publications

(212 citation statements)

References 38 publications

(46 reference statements)

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“…In the case of vector fields in dimension 3, the main result has been obtained by Morales, Pacifico and Pujals in 2004 [49]. Let be a flow on a manifold , preserving a compact set Λ.…”

Section: Ergodic Theory Of the Lorenz Attractormentioning

confidence: 88%

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The Lorenz Attractor, a Paradigm for Chaos

Ghys

2013

Progress in Mathematical Physics

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Abstract. It is very unusual for a mathematical or physical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz's butterfly effect: "does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" A tiny cause can generate big consequences! Mathematicians (and non mathematicians) have known this fact for a long time! Can one adequately summarize chaos theory is such a simple minded way? In this review paper, I would like first of all to sketch some of the main steps in the historical development of the concept of chaos in dynamical systems, from the mathematical point of view. Then, I would like to present the present status of the Lorenz attractor in the panorama of the theory, as we see it Today.

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“…In the case of vector fields in dimension 3, the main result has been obtained by Morales, Pacifico and Pujals in 2004 [49]. Let be a flow on a manifold , preserving a compact set Λ.…”

Section: Ergodic Theory Of the Lorenz Attractormentioning

confidence: 88%

“…The main theorem in [49] states that if a compact invariant set Λ of a three-dimensional flow is robustly transitive, then:…”

Section: Ergodic Theory Of the Lorenz Attractormentioning

confidence: 99%

The Lorenz Attractor, a Paradigm for Chaos

Ghys

2013

Progress in Mathematical Physics

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“…It has been further conjectured ( [15,Conjecture 4]) that the last situation above can be replaced by a singular hyperbolic set (see [12] for the definition).…”

Section: An Applicationmentioning

confidence: 99%

Hamiltonian suspension of perturbed Poincaré sections and an application

Bessa

Dias²

2014

Math. Proc. Camb. Phil. Soc.

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Abstract. We construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincaré map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension 2d. As an application, using known properties of area-preserving maps, we prove that for any Hamiltonian defined on a symplectic 4-manifold M and any point p ∈ M , there exists a C 2 -close Hamiltonian whose regular energy surface through p is either Anosov or contains a homoclinic tangency.

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“…As referências principais são [2,14,18,20,23,26,28] e outras que serviram na complementação da teoria foram [7,12,19,22].…”

Section: Fluxosunclassified

“…Atratores e repulsores são conjuntos compactos e invariantes pelo fluxo. Para maiores detalhes sobre as propriedades de atratores veja [2,13,19].…”

Section: Definições E Propriedadesunclassified

Sistemas quase-Anosov

Aguirre¹

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AgradecimentosAgradeço principalmente a Deus pela saúde, pelas vitórias conquistadas e bem-estar que nunca me ha faltado.Ao meu orientador, Prof. Dr. Jean Venato Santos pela sua paciência e disponibilidade. Pela ajuda em esclarecer minhas dúvidas da pesquisa, por seus conselhos e discussões construtivas sobre a pesquisa e demais assuntos. Também agradeço a confiança, amizade e tempo que teve comigo.A AbstractIn 1970, Hirsch [10] conjectured that given a diffeomorphism f : M → M in a differentiable manifold M, if N ⊂ M is a compact invariant submanifold with a hyperbolic structure as a subset of M , then f restricted to N would be a Anosov diffeomorphism. In this work we present a counterexample to this conjecture published in 1976 by Franks and Robinson [8]. Next we present a result of Zeghib [30] showing that (φ, M ) is an Anosov system (flow or diffeomorphism) with splitting property, given a closed invariant submanifold N of M , then (φ, N ) is a transitive Anosov system.

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Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers

Cited by 148 publications

References 38 publications

The Lorenz Attractor, a Paradigm for Chaos

The Lorenz Attractor, a Paradigm for Chaos

Hamiltonian suspension of perturbed Poincaré sections and an application

Sistemas quase-Anosov

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