Pseudo-orbit shadowing in the family of tent maps

Coven, Ethan M.; Kan, Ittai; Yorke, James A.

doi:10.1090/s0002-9947-1988-0946440-2

Cited by 97 publications

(59 citation statements)

References 2 publications

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“…Moreover, it may be seen that is a non-wandering dynamical system with shadowing. A particular example here is the full tent map f (x) = 1 − |1 − 2x| or various other maps from the family of tent maps [15].…”

Section: Theorem 54 Let (X F ) Be a Non-wandering And Topologicallymentioning

confidence: 99%

Shadowing, entropy and minimal subsystems

Moothathu

Oprocha

2013

Monatsh Math

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We consider non-wandering dynamical systems having the shadowing property, mainly in the presence of sensitivity or transitivity, and investigate how closely such systems resemble the shift dynamical system in the richness of various types of minimal subsystems. In our excavation, we do discover regularly recurrent points, sensitive almost 1-1 extensions of odometers, minimal systems with positive topological entropy, etc. We also show that transitive semi-distal systems with shadowing are in fact minimal equicontinuous systems (hence with zero entropy) and, in contrast to systems with shadowing, the entropy points do not have to be densely distributed in transitive systems.

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Section: Theorem 54 Let (X F ) Be a Non-wandering And Topologicallymentioning

confidence: 99%

Shadowing, entropy and minimal subsystems

Moothathu

Oprocha

2013

Monatsh Math

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“…Furthermore, they have been the subject of many research articles [12,17,28,30], often in relation to their ω-limit sets [6,21,22,23,24]. In this paper we make several important observations about the behaviour of tent maps, allowing us to prove new results about the nature of their limit sets in relation to certain well-known dynamical properties.…”

Section: Introductionmentioning

confidence: 95%

“…These maps do not have shadowing as the critical point is not recurrent [17]. Thus we suggest the following revision of Conjecture 1.1: We believe that this conjecture would be of interest, despite the fact that we know it holds for limit shadowing rather than shadowing, because the two properties are not equivalent [29], and shadowing is simple to test for [17].…”

Section: Introductionmentioning

confidence: 99%

On the ω-limit sets of tent maps

2012

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Abstract. For a continuous map f on a compact metric space (X, d), a set D ⊂ X is internally chain transitive if for every x, y ∈ D and every δ > 0 there is a sequence of points x = x 0 , x 1 , . . . , x n = y such that d(f (x i ), x i+1 ) < δ for 0 ≤ i < n. It is known that every ω-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D ⊂ X is internally chain transitive if and only if D is an ω-limit set for some point in X, and that the same is also true for the full tent map T 2 : [0, 1] → [0, 1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.

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“…Furthermore, if the above example lies on a space M 1 , and on another space M 2 there is a hyperbolic attractor, then the attractor for the product system is chaotic and has the shadowing property, but the attractor is not hyperbolic. For a simpler example, Coven, Kan and Yorke [9] have shown that tent maps are shadowable for all parameter values (i.e. , the absolute value of the slopes) between 1 and 2 except for a set having zero Lebesgue measure.…”

Section: Introductionmentioning

confidence: 99%

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

Yuan

Yorke

1999

Proc. Amer. Math. Soc.

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Abstract. We consider a class of nonhyperbolic systems, for which there are two fixed points in an attractor having a dense trajectory; the unstable manifold of one has dimension one and the other's is two dimensional. Under the condition that there exists a direction which is more expanding than other directions, we show that such attractors are nonshadowable. Using this theorem, we prove that there is an open set of diffeomorphisms (in the C rtopology, r > 1) for which every point is absolutely nonshadowable, i.e., there exists > 0 such that, for every δ > 0, almost every δ-pseudo trajectory starting from this point is -nonshadowable.

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Pseudo-orbit shadowing in the family of tent maps

Cited by 97 publications

References 2 publications

Shadowing, entropy and minimal subsystems

Shadowing, entropy and minimal subsystems

On the ω-limit sets of tent maps

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

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