Partial hyperbolicity and robust transitivity

Díaz, Lorenzo J.; Pujals, Enrique R.; Ures, Raúl

doi:10.1007/bf02392945

Cited by 100 publications

(72 citation statements)

References 23 publications

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“…However, in the presence of singularities, this result cannot be applied: a singularity is an obstruction to consider the flow as the suspension of a 2-diffeomorphism. On the other hand, for diffeomorphisms on 3-manifolds it has recently been proved that any robust transitive set is partially hyperbolic [8]. Again, this result cannot be applied to the time-one diffeomorphism X 1 to prove Theorem C: if Λ is a saddle-type periodic point of X then Λ is a robust transitive set for X, but not necessarily a robust transitive set for X 1 .…”

Section: Related Results and Commentsmentioning

confidence: 97%

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

2004

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Inspired by Lorenz' remarkable chaotic flow, we describe in this paper the structure of all C 1 robust transitive sets with singularities for flows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C 1 robust attractor with singularities for flows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the flow, and has positive Lyapunov exponent at every orbit, showing that any C 1 robust attractor resembles a geometric Lorenz attractor.

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Section: Related Results and Commentsmentioning

confidence: 97%

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

2004

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“…The results of C. Morales, M.J. Pacifico and E. Pujals [25] provide a unified framework for robust strange attractors in dimension 3 of which the Lorenz attractor is a particular case. See also the paper [9] by L. Díaz, E. Pujals and R. Ures for related results about discrete-time systems. The topological classification of the Lorenz attractors (for different parameter values) can be found in the paper [29] by D. Rand.…”

Section: The Lorenz Attractor Near the Hopf Bifurcationmentioning

confidence: 98%

Global topological properties of the Hopf bifurcation

Sanjurjo

2007

Journal of Differential Equations

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We study the homotopical and homological properties of the attractors evolving from a generalized Hopf bifurcation. We consider the Lorenz equations for parameter values near the Hopf bifurcation and study a natural Morse decomposition of the global attractor, calculating theČech homotopy type of the Lorenz attractor, the shape indexes of the Morse sets and the Morse equation of the decomposition.

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“…For instance, Mañé [8] proved that if a diffeomorphism on two-dimensional C ∞ manifolds is robustly transitive, then it is hyperbolic, and Díaz et al [5] proved that if a diffeomorphism on threedimensional C ∞ manifold is robustly transitive then it is partially hyperbolic. And, in [1], the authors proved that for any dimensional C ∞ manifolds, if a diffeomorphism is robustly transitive, then it admits a dominated splitting.…”

Section: Introductionmentioning

confidence: 99%