On the shadowing and limit shadowing properties

Kawaguchi, Noriaki

doi:10.4064/fm552-4-2019

Cited by 4 publications

(4 citation statements)

References 32 publications

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“…• f has the limit-shadowing property, but its restriction to CR(f ) does not have the limit-shadowing property, • f exhibits duDC2 and dDC1 but does not exhibit gDC2. Note that the third property gives an answer to a question in [4].…”

Section: Examplesmentioning

confidence: 99%

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Generic and dense distributional chaos with shadowing

Kawaguchi¹

2021

Preprint

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For continuous self-maps of compact metric spaces, we consider the notions of generic and dense chaos introduced by Lasota and Snoha and their variations for the distributional chaos, under the assumption of shadowing. We give some equivalent properties to generic uniform (distributional) chaos in terms of chains and prove their equivalence to generic (distributional) chaos under the shadowing. Several examples illustrating the main results are also given.

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Section: Examplesmentioning

confidence: 99%

“…The third property gives an answer to a question in [4]. Example 4.4 provides a continuous map with the shadowing property exhibiting guDC1.…”

Section: Introductionmentioning

confidence: 99%

Generic and dense distributional chaos with shadowing

Kawaguchi¹

2021

Preprint

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“…The end of this proof proves that if a minimal homeomorphism satisfies the two-sided limit shadowing property with a gap, then the space is finite and a single periodic orbit. In particular, the adding machines do not satisfy the two-sided limit shadowing property with a gap, even though they satisfy the limit shadowing property (see [13]). They satisfy even the orbital two-sided limit shadowing property, as defined in [18].…”

Section: Singular Suspensionsmentioning

confidence: 99%

Suspensions of homeomorphisms with the two-sided limit shadowing property

2019

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In this paper we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyze some of the results known for the case of homeomorphisms in the case of continuous flows and observe that some differences appear in this scenario. We prove that the suspension flow of a homeomorphism satisfying the two-sided limit shadowing property also satisfies it. This gives a lot of examples of flows satisfying this property, however it enlighten an important difference between the case of flows and homeomorphisms: there are flows satisfying the two-sided limit shadowing property that are not topologically mixing, while homeomorphisms satifying the two-sided limit shadowing property satisfy even the specification property. There are no homeomorphisms on the circle satisfying the two-sided limit shadowing property but we exhibit examples of flows on the circle satisfying it. It can happen that a suspension flow has the two-sided limit shadowing property but the base homeomorphism does not, though it is proved that it must satisfy a strictly weaker property called two-sided limit shadowing with a gap (as in [10]). We define a similar notion of two-sided limit shadowing with a gap for flows and prove that these notions are actually equivalent in the case of flows. Finally, we prove that singular suspension flows (in the sense of Komuro [14]) do not satisfy the two-sided limit shadowing property.

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“…The theory of shadowing in dynamical systems is a rapidly developing branch of modern global theory of dynamical systems. There are various notions of shadowing for dynamical systems (homeomorphisms or flows) on compact metric spaces, and a lot of interesting results of dynamical systems with various type of shadowing were investigated in many papers (e.g., see [2,4,6,7,9,11,13]).…”

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confidence: 99%

Flows with the weak two-sided limit shadowing property

Lee¹,

Nguyen²

2021

DCDS

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In this paper we study the weak two-sided limit shadowing for flows on a compact metric space which is different with the usual shadowing, two-sided limit shadowing and L-shadowing, and characterize the weak twosided limit shadowing flows from the pointwise and measurable viewpoints. Moreover, we prove that if a flow φ has the weak two-sided limit shadowing property on its chain recurrent CR(φ) then the set CR(φ) is decomposed by a finite number of closed invariant sets on which φ is topologically transitive and has the two-sided limit shadowing property.

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On the shadowing and limit shadowing properties

Cited by 4 publications

References 32 publications

Generic and dense distributional chaos with shadowing

Generic and dense distributional chaos with shadowing

Suspensions of homeomorphisms with the two-sided limit shadowing property

Flows with the weak two-sided limit shadowing property

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