Homoclinic classes for generic C^1 vector fields

Carballo, C. M.; Morales, C. A.; Pacífico, Maria José

doi:10.1017/s0143385702001116

Cited by 43 publications

(59 citation statements)

References 16 publications

(26 reference statements)

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“…The converse, although false in general, is true for a residual subset of C 1 vector fields [3]. We derive a sufficient condition for the validity of the converse to this result inspired by the following well known property of hyperbolic attractors [31]: If Λ is a hyperbolic attractor of a vector field X, then there is an isolating block U of Λ and…”

Section: Attractors and Isolated Sets For C 1 Flowsmentioning

confidence: 99%

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: Attractors and Isolated Sets For C 1 Flowsmentioning

confidence: 99%

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

2004

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“…The results in [Ku,Sm 1 ,Pu,CMP] give a residual subset G of Diff 1 (M) of diffeomorphisms f verifying the following properties:…”

Section: Summary Of Generic Properties Of C 1 -Diffeomorphismsmentioning

confidence: 99%

Periodic points and homoclinic classes

Abdenur¹,

Bonatti²,

Crovisier³

et al. 2006

Ergod. Th. Dynam. Sys.

103

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We prove that there is a residual subset I of Diff 1 (M) such that any homoclinic class of a diffeomorphism f ∈ I having saddles of indices α and β contains a dense subset of saddles of index τ for every τ ∈ [α, β] ∩ N. We also derive some consequences from this result about the Lyapunov exponents of periodic points and the sort of bifurcations inside homoclinic classes of generic diffeomorphisms.

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“…Thus in the hyperbolic case different homoclinic classes are disjoint. Moreover, these properties were proved to be true for a residual subset of C 1 diffeomorphisms on any n-dimensional manifold [9].…”

Section: Introductionmentioning

confidence: 91%

“…In [2] it is proved that for generic C 1 diffeomorphisms, the elementary pieces are the chain recurrent classes, and when one of these sets contains a periodic point p it coincides with the homoclinic class of p. Moreover, [9] together with the Closing Lemma of [32], give that the homoclinic classes constitute a partition of a dense part of the limit set of generic diffeomorphisms, and [10] establishes that chain recurrent sets of generic diffeomorphisms are Hausdorff limits of homoclinic classes. All these results evidence the importance of understanding the dynamics restricted to homoclinic classes.…”

Section: Introductionmentioning

confidence: 98%