Stability of singular horseshoes

Labarca, Rafael; Pacífico, Maria José

doi:10.1016/0040-9383(86)90048-0

Cited by 48 publications

(49 citation statements)

References 6 publications

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“…We also point out that a transitive singular-hyperbolic set is not necessarily a robust transitive set, even in the case that the set is an attractor; see [17] and [27]. So, the converse of our results requires extra conditions that are yet unknown.…”

Section: Related Results and Commentsmentioning

confidence: 88%

“…It is false as well in the context of boundary-preserving vector fields on 3-manifolds with boundary [17]. The converse to Theorem A is also not true: proper attractors (or repellers) with singularities are not necessarily robust transitive, even if their periodic points and singularities are hyperbolic in a robust way.…”

Section: Theorem a A Robust Transitive Set Containing Singularities mentioning

confidence: 88%

“…These examples are also C 1 robust transitive. On the other hand, the singular horseshoe [17] and the example in [27] are neither C 1 robust transitive nor C 1 robust periodic. These examples motivate the question whether all C 1 robust transitive sets for vector fields are C 1 robust periodic.…”

Section: Attractors and Isolated Sets For C 1 Flowsmentioning

confidence: 99%

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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: 88%

Section: Theorem a A Robust Transitive Set Containing Singularities mentioning

confidence: 88%

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

2004

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“…In the absence of equilibria, robustness implies uniform hyperbolicity. The first examples of singular hyperbolic sets included the Lorenz attractor [26,45] and its geometric models [15,1,16,49], and the singular-horseshoe [24], besides the uniformly hyperbolic sets themselves. Many other examples have recently been found, including attractors arising from certain resonant double homoclinic loops [38] or from certain singular cycles [33], and certain models across the boundary of uniform hyperbolicity [32].…”

Section: Introductionmentioning

confidence: 99%

Singular-hyperbolic attractors are chaotic

Araújo

Pacífico

Pujals

et al. 2008

Trans. Amer. Math. Soc.

120

195

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Abstract. We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-RuelleBowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a u-Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strongunstable direction.This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are u-Gibbs states.

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“…This paper is motivated by [14], where C 1 structural stability for vector fields on manifolds with boundary, called Singular Horseshoe, was proved. To do this is necessary to prove C 1 structural stability for the expanding one-dimensional maps modeling the dynamic.…”

Section: Introductionmentioning

confidence: 99%

One-dimensional contracting singular horseshoe

Carrasco-Olivera

Morales

Martín

2010

Proc. Amer. Math. Soc.

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Abstract. In this paper we prove some kind of structural stability defined as usual but restricted to a certain subset of one-dimensional maps coming from first return maps associated to singular cycles for vector fields in manifolds with boundary. The motivation is the stability of the Singular Horseshoes introduced by Labarca and Pacifico where an expanding condition on the singularity holds. Here we obtain analogous result but under a contracting condition.

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Stability of singular horseshoes

Cited by 48 publications

References 6 publications

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

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

Singular-hyperbolic attractors are chaotic

One-dimensional contracting singular horseshoe

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