On persistently positively expansive maps

Arbieto, Alexander

doi:10.1590/s0001-37652010000200002

Cited by 5 publications

(2 citation statements)

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“…Mañé, in [9] proved that, in the invertible case, if any diffeomorphism in a C 1 -neighborhood of some diffeomorphism is expansive then this neighborhood is formed by Axiom A diffeomorphisms. In [1], the author shows that for non-invertible maps, if any local diffeomorphism in a C 1 -neighborhood of some local diffeomorphism is positively expansive then this neighborhood is formed by expanding maps.…”

Section: Introductionmentioning

confidence: 99%

Periodic orbits and expansiveness

Arbieto

2010

Math. Z.

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Section: Introductionmentioning

confidence: 99%

Periodic orbits and expansiveness

Arbieto

2010

Math. Z.

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“…[22,31,51,18]) inspired the works of many authors to approach such dichotomy in the space of C 1 -diffeomorphisms concerning other important dynamical properties that are not necessarily C 1 -open, namely, expansiveness, shadowing or specification properties. In [3,35,40,41] it was proved that the C 1 -interior of the set of all C 1 -diffeomorphisms satisfying any of these properties is contained in the set of uniformly hyperbolic diffeomorphisms.…”

Section: Introductionmentioning

confidence: 99%

Specification and partial hyperbolicity for flows

2015

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We prove that if a flow exhibits a partially hyperbolic attractor Λ with splitting T Λ M = E s ⊕ E c and two periodic saddles with different indices such that the stable index of one of them coincides with the dimension of E s then it does not satisfy the specification property. In particular, every singular-hyperbolic attractor with the specification property is hyperbolic. As an application, we prove that no Lorenz attractor satisfies the specification property.1 for t ≥ 0. Now, take a small t 2 > 0 such thatz)) ≤ 2L 0 ε} and t 0 = inf I. By (3.21), (3.25) and (3.26) we have T ∈ I. Assume that t 0 > 0. Sinceby (3.21) and (3.26) we have t 0 − t 2 ∈ I, which is a contradiction. Thus 0 = inf I. ThereforeThe proposition follows taking L = 3L 0 . Lemma 3.10. Let p ∈ Λ be a hyperbolic critical element with dim W ss (p) = dim E s p and U be a subdisk of W u (p) with U ⊂ W u (p). Then there exists µ > 0 such that A(U ) := z∈U F s µ (z) is homeomorphic to U × [−µ, µ] dim E s .

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