An open set of maps for which every point is absolutely nonshadowable

Yuan, Guo‐Cheng; Yorke, James A.

doi:10.1090/s0002-9939-99-05038-8

Cited by 40 publications

(16 citation statements)

References 21 publications

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“…The partially hyperbolic assumption prevents this pseudo-orbit from being shadowed by a true g-orbit (this is proved in [2,Lemma 3.12], see also [12,44] where the similar arguments are used). Therefore, N is an open set of diffeomorphisms which do not satisfy the shadowing property.…”

Section: Lemma 66mentioning

confidence: 98%

“…Recall that Theorem 1.17 claims that every diffeomorphism with a co-index 1 cycle is in the closure of an open set of Diff 1 (M ) of diffeomorphism which do not satisfy the shadowing property. The proof of Theorem 1.17 follows using the arguments in [2, Theorem 1] (in its turn, these arguments are an adaptation of the ones in [12,44]). Let us sketch these arguments.…”

Section: The Definition Of the Perturbation And The Choice Ofmentioning

confidence: 99%

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Robust Heterodimensional Cycles and $C^1$-Generic Dynamics

Bonatti¹,

Díaz²

2007

JMJ

117

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A diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets Λ and Σ having different indices (dimension of the unstable bundle) such that the unstable manifold of Λ meets the stable one of Σ and vice versa. This cycle has co-index 1 if index(Λ) = index(Σ) ± 1. This cycle is robust if, for every g close to f , the continuations of Λ and Σ for g have a heterodimensional cycle.We prove that any co-index 1 heterodimensional cycle associated with a pair of hyperbolic saddles generates C 1 -robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles.We also derive some consequences from this result for C 1 -generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle. 471Question 1.2. Let M be closed manifold. Does it exist a C 1 -open and dense subset O ⊂ Diff 1 (M ) such that every f ∈ O either verifies the Axiom A and the no-cycles condition or has a C 1 -robust heterodimensional cycle?Note that a positive answer to this question implies the C 1 -density of hyperbolic surface diffeomorphisms. See the discussion in § 1.3 about the Smale density conjecture. We will see that Theorem 1.14 gives a partial positive answer to this question for the so-called tame diffeomorphisms (diffeomorphisms finitely many homoclinic classes, see the preciseThe examples by Abraham-Smale of non-Axiom A diffeomorphisms involves a hyperbolic set Γ whose unstable manifold has dimension strictly greater than the dimension of its unstable bundle. Note that a normally hyperbolic extension of transitive Anosov diffeomorphisms on a torus T 2 gives an example of this configuration.The construction in [8] gives a slightly different mechanism for constructing non-Axiom A diffeomorphisms and robust heterodimensional cycles, based on the notion of blender. Roughly speaking, a blender is a hyperbolic set whose embedding in the ambient manifold verifies some specific geometric properties, whose effect is that, as in the Abraham-Smale example, the unstable manifold of a blender looks like a manifold of higher dimension. We review the construction and main properties of blenders in § 4.1.3. See also [15, Chapter 6.1] for a discussion of this notion.One of the goals of this paper is to show that blenders (and as a consequence robust heterodimensional cycles) appear in a natural way in the unfolding of heterodimensional cycles associated with two saddles. Definition 1.3 (heterodimensional cycle and co-index 1 cycle).A diffeomorphism f has a heterodimensional cycle (see Figure 1) associated with two hyperbolic periodic saddles P and Q of f if the saddles P and Q have different indices, the stable manifold of the orbit of P meets the unstable manifold of...

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Section: Lemma 66mentioning

confidence: 98%

Section: The Definition Of the Perturbation And The Choice Ofmentioning

confidence: 99%

Robust Heterodimensional Cycles and $C^1$-Generic Dynamics

Bonatti¹,

Díaz²

2007

JMJ

117

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“…The FE property implies shadowing fails, as was established by Dawson et al [16]. Homogeneous chaotic systems can have the shadowing property but hetero-chaotic systems cannot, as shown for UDV in [25][26][27].…”

Section: Hetero-chaos Connects Many Phenomena Like Fluctuating Ementioning

confidence: 99%

Low-dimensional paradigms for high-dimensional hetero-chaos

Saiki

Sanjuán

Yorke

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time while typical trajectories wander throughout the attractor. Such an attractor is "hetero-chaotic" (i.e. it has heterogeneous chaos) if furthermore arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions. This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight to real high-dimensional phenomena.Prediction and simulation for chaotic systems occur throughout science. Predictability is more difficult when the "chaotic attractor" is heterogeneous, i.e. if different regions of the chaotic attractor are unstable in more directions than in others. More precisely, when arbitrarily close to each point of the attractor there are different periodic points with different unstable dimensions, we say the chaos is heterogeneous and we call it hetero-chaos. Simple illustrative models of hetero-chaos have been lacking in the literature, and here we present the simplest examples we have found.

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“…A first observation is that one cannot expect to recover the shadowing property, at least in its full strength, for general partially hyperbolic systems. In fact, it was observed in [8] that the shadowing property is not verified (not even generically) for partially hyperbolic diffeomorphisms which are robustly transitive (see also [1,25] for examples of some large classes of non-uniformly hyperbolic systems that do not satisfy the shadowing property). On the other hand, as we are going to see in the sequel, it is possible to get a weaker version of the shadowing property called quasi-shadowing.…”

Section: Introductionmentioning

confidence: 99%

Quasi-shadowing for partially hyperbolic dynamics on Banach spaces

Backes,

Dragicevic

2020

Preprint

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A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points (xn) n∈Z such that xn+1 is obtained from the image of xn by moving it by a small factor in the central direction. In the present paper, we prove that a small nonlinear perturbation of a partially dichotomic sequence of (not necessarily invertible) linear operators acting on an arbitrary Banach space has the quasi-shadowing property. We also get obtain a continuous time version of this result. As an application of our main result, we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction.

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An open set of maps for which every point is absolutely nonshadowable

Cited by 40 publications

References 21 publications

Robust Heterodimensional Cycles and $C^1$-Generic Dynamics

Robust Heterodimensional Cycles and $C^1$-Generic Dynamics

Low-dimensional paradigms for high-dimensional hetero-chaos

Quasi-shadowing for partially hyperbolic dynamics on Banach spaces

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