Expansive one-parameter flows

Bowen, Rufus; Walters, Peter

doi:10.1016/0022-0396(72)90013-7

Cited by 246 publications

(269 citation statements)

References 12 publications

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“…Using Lemma 3.2, we conclude that dist(X s+τ −t (x), X h(s+τ −t) (y)) ≥ cρ > δ n which contradicts (10). This contradiction reduces the proof of Theorem A to that of Theorem 3.1.…”

Section: Proof Of Expansivenessmentioning

confidence: 65%

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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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Section: Proof Of Expansivenessmentioning

confidence: 65%

“…We say that the sub-bundle E cu A stronger notion of expansiveness has been proposed by Bowen-Walters [10]. In it one considers continuous maps h : R → R with h(0) = 0, instead.…”

Section: Definition 11 An Attractor Is a Transitive Attracting Setmentioning

confidence: 99%

Singular-hyperbolic attractors are chaotic

Araújo

Pacífico

Pujals

et al. 2008

Trans. Amer. Math. Soc.

120

195

View full text Add to dashboard Cite

show abstract

“…We can now clearly extend η(t) to the interval [0, 2] in such a way that η is C ∞ and η(t) = 1 for t ∈ [1,2]. To finish the proof of the lemma, we set α i = (i + 1) −1 in the above construction of η(t) and let ω(x) = η(|x|) on B n (2).…”

Section: Lemma 31 For a Given Sequence Of Positive Numbersmentioning

confidence: 99%

Topological entropies of equivalent smooth flows

Sun¹,

Young²,

Zhou³

2008

Trans. Amer. Math. Soc.

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“…The following definition is the notion of expansive vector field, introduced by Bowen and Walters, in [6].…”

Section: Introduction and Results' Statementmentioning

confidence: 99%

Shadowing, Expansiveness and Stability of Divergence-Free Vector Fields

Ferreira¹

2014

Bulletin of the Korean Mathematical Society

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Abstract. Let X be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions:• X is a divergence-free vector field satisfying the shadowing property.• X is a divergence-free vector field satisfying the Lipschitz shadowing property.• X is an expansive divergence-free vector field.• X has no singularities and is Anosov.

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Expansive one-parameter flows

Cited by 246 publications

References 12 publications

Singular-hyperbolic attractors are chaotic

Singular-hyperbolic attractors are chaotic

Topological entropies of equivalent smooth flows

Shadowing, Expansiveness and Stability of Divergence-Free Vector Fields

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