Expansivity and unique shadowing

Good, Chris; Macı́as, Sergio; Meddaugh, Jonathan; Mitchell, Joel; Thomas, Joe

doi:10.1090/proc/15204

Cited by 6 publications

(5 citation statements)

References 47 publications

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“…This property is strictly weaker than the two-sided limit shadowing property studied by various authors (e.g. [10][11][12]20,35]). An irrational rotation of the circle will have two-sided orbital limit shadowing but not two-sided limit shadowing.…”

Section: It Is Easily Observed By Looking Atmentioning

confidence: 53%

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When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Mitchell

2022

Topology and its Applications

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Section: It Is Easily Observed By Looking Atmentioning

confidence: 53%

“…i ∈ Z). In [20], we show that shadowing implies backward and two-sided shadowing, whilst all three properties are equivalent for surjective maps. Using this, in [23], we showed that shadowing is sufficient for the property in Question 1.2.…”

Section: Preliminariesmentioning

confidence: 86%

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Mitchell

2022

Topology and its Applications

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“…In [5] the first author et al showed that an expansive map has shadowing if and only if it has s-limit shadowing. We extended that result in [22] to show that an expansive map has shadowing if and only if it has two-sided s-limit shadowing. Combining this result with Theorem 4.8, we immediately obtain the following.…”

Section: Shadowing Ict and α Fmentioning

confidence: 80%

Shadowing, internal chain transitivity and α-limit sets

Good

Meddaugh

Mitchell

2020

Journal of Mathematical Analysis and Applications

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“…Similarly it has two-sided shadowing if for any ε > 0 there exists δ > 0 such that for any two-sided δ-pseudo-orbit x i i∈Z there exists a full trajectory z i i∈Z such that d(x i , z i ) < ε for all i ∈ Z. In [20], we show that shadowing implies backward and two-sided shadowing, whilst all three properties are equivalent for surjective maps. Using this, in [23], we showed that shadowing is sufficient for P a .…”

Section: Preliminariesmentioning

confidence: 81%

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Mitchell¹

2020

Preprint

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Let f : X → X be a continuous map on a compact metric space X and let α f , ω f and ICT f denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of ICT f is equal to (resp. may be approximated by) the α-limit set and the ω-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.

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Expansivity and unique shadowing

Cited by 6 publications

References 47 publications

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Shadowing, internal chain transitivity and α-limit sets

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

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