Maximal transitive sets with singularities for genericC 1 vector fields

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

doi:10.1007/bf01241631

Cited by 13 publications

(12 citation statements)

References 9 publications

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“…In other words, there exists |τ | ≤ ε such that X h(s) (y) ∈ W ss ε (X s+τ (x)). Hypothesis (11) implies that X h(s) (y) = X s+τ (x). Since strong-stable manifolds are expanded under backward iteration, there exists θ > 0 maximum such that…”

Section: Proof Of Expansivenessmentioning

confidence: 99%

Singular-hyperbolic attractors are chaotic

Araújo

Pacífico

Pujals

et al. 2008

Trans. Amer. Math. Soc.

121

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: 99%

Singular-hyperbolic attractors are chaotic

Araújo

Pacífico

Pujals

et al. 2008

Trans. Amer. Math. Soc.

121

195

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“…Any forward orbit for ϕ L with an initial point in T L can not escape from T L . The invariant set t≥0 ϕ L (T L , t) for X L does not have any continuous hyperbolic splitting at 0, but it belongs to an essential class called singular hyperbolic, which is studied extensively from various approaches by Morales, Pacifico and others, see for details [3,4,11,12,13,14] . Now, we introduce the notion of PSSP for Lorenz flows.…”

Section: Preliminariesmentioning

confidence: 99%

Parameter-shifted shadowing property of Lozi maps

Kiriki

Soma

2007

Dynamical Systems

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“…for X L does not have any continuous hyperbolic splitting at 0, but it belongs to an essential class called singular hyperbolic, which is studied extensively from various approaches by Morales, Pacifico and others; see for details [3,4,11,12,13,14]. (i) A sequence {x n } n≥0 in T ψ with x 0 ∈ Σ is a (δ, τ )-pseudo-orbit for the flow ψ if there exists a sequence {τ n } n≥0 such that, for any n ≥ 0,…”

Section: Theorem 23 (Pssp For Lorenz Planar Maps) Any Lorenz Map L mentioning

confidence: 99%