On certain hyperbolic sets

Аносов, Дмитрий Викторович

doi:10.1134/s0001434610050020

Cited by 6 publications

(10 citation statements)

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“…In particular, there were formulated some "intrinsic" conditions that guarantee lack of local premaximality appropriate to these examples. A more recent general theorem [3,5] overlaps with observations contained in [4] that the lack of local premaximality in these examples is connected with "intrinsic" peculiarities of the behavior of trajectories. But it is possible that these peculiarities are to some extent interesting as such.…”

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confidence: 67%

“…8 With a clause that similar arguments were briefly hinted at in [2,3]. 9 The main example from [8] (with some simple modification) was discussed in [4]. In particular, there were formulated some "intrinsic" conditions that guarantee lack of local premaximality appropriate to these examples.…”

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confidence: 98%

“…Therefore we need only that functions fi : Ui → Mi with necessary characteristics be defined in some neighborhoods Ui of Fi. However, the full formulation of all these characteristics is rather cumbersome, though the "semilocal" statement is formulated as one that is concerned with the self-diffeomorphism of the whole phase manifold in the theory of smooth dynamical systems 4 It recalls a weak isomorphism in the ergodic theory. But is it important?…”

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confidence: 99%

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On Trajectories Entirely Situated Near a Hyperbolic Set

Anosov

2014

J Math Sci

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Let I1 be a set of points such that their trajectories under a diffeomorphism f1 are entirely close enough to a hyperbolic set F1 of this diffeomorphism. Then it is proved that the structure of I1 and restriction f1|I 1 ("motion in I1") are essentially defined (up to an equivariant homeomorphism) by "internal dynamics" in F1, i.e., by the restriction f1|F 1 . (In more detail: the equivariant homeomorphism g1 of the set F1 on the hyperbolic set F2 of the second diffeomorphism f2 (probably, acting on another manifold M2) is extendable to an equivariant homeomorphic embedding I1 → M2. The image of the imbedding contains all the trajectories f2 close enough to F2.) 1. Before formulating the results, let us turn our attention to notation and terminology that we use further.Let A be a subset of the set M. Let us denote by j A an identical embedding of A in M, i.e., if x ∈ A, then j A x is the same element x, but it is considered as an element of M. (This notation will be used in cases where we talk mostly about the same M. Therefore, we do not consider M in the notation.)As before, let A ⊂ M and let a bijective transformation f : M → M act on the set M. Let us denote I(A) := ∞ −∞

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confidence: 98%

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confidence: 99%

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On Trajectories Entirely Situated Near a Hyperbolic Set

Anosov

2014

J Math Sci

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“…Before we proceed to the proof of Theorems 1.5 and 1.6 let us review results by Anosov on premaximality [2,3,4,5,6]. As was mentioned in Section 1, Theorem 1.4 implies that premaximality is an intrinsic property.…”

Section: Premaximalitymentioning

confidence: 94%

“…In Section 3 we prove Theorem 1.2. In Section 4 we review the results of Anosov in [2,3,4,5,6] and prove Theorems, 1.5, 1.6.…”

Section: Introductionmentioning

confidence: 99%

Nonlocally maximal and premaximal hyperbolic sets

Fisher¹,

Petty²,

Tikhomirov³

2017

Modern Theory of Dynamical Systems

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We prove that for any closed manifold of dimension 3 or greater that there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a hyperbolic set is an intrinsic property for premaximality. Lastly, we review some results due to Anosov that concern premaximality. 2000 Mathematics Subject Classification. 37D20, 37D05, 37C05.

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Intrinsic Character of One Property of Hyperbolic Sets

Anosov

2010

J Dyn Control Syst

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On certain hyperbolic sets

Cited by 6 publications

References 2 publications

On Trajectories Entirely Situated Near a Hyperbolic Set

On Trajectories Entirely Situated Near a Hyperbolic Set

Nonlocally maximal and premaximal hyperbolic sets

Intrinsic Character of One Property of Hyperbolic Sets

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