N-expansive homeomorphisms with the shadowing property

Carvalho, Bernardo; Cordeiro, Welington

doi:10.1016/j.jde.2016.06.003

Cited by 46 publications

(40 citation statements)

References 12 publications

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“…Several generalizations of expansivity were considered before: the n-expansive systems in [5,8,16,17,32,?LZ], finite expansiveness in [16], countable and measure expansivity in [5,7,19], cw-expansive homeomorphisms in [23,24] and entropy expansiveness in [11,34] (among others). The L-shadowing property was defined in [17] as an attempt to link shadowing and Morales' n-expansivity [32].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Beyond topological hyperbolicity: The L-shadowing property

Artigue

Carvalho

Cordeiro

et al. 2020

Journal of Differential Equations

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In this paper we further explore the L-shadowing property defined in [17] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy this property. Homeomorphisms satisfying the L-shadowing property have a spectral decomposition where the basic sets are either expansive or contain arbitrarily small topological semi-horseshoes (periodic sets where the restriction is semiconjugate to a shift). To this end, we characterize the L-shadowing property using local stable and unstable sets and the classical shadowing property. We exhibit homeomorphisms with the Lshadowing property and arbitrarily small topological semi-horseshoes without periodic points. At the end, we show that positive finite-expansivity jointly with the shadowing property imply that the space is finite.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Beyond topological hyperbolicity: The L-shadowing property

Artigue

Carvalho

Cordeiro

et al. 2020

Journal of Differential Equations

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“…In topological dynamics, an important place is given to the shadowing theory, where many variants of pseudo-orbit tracing properties are discussed, mainly considering different notions of pseudo-orbits and shadowing points. Among them there is the limit shadowing property which has been given much attention recently (see [5], [6], [7], [8], [10], [11], [15], [16], [18], [19] and others). It deals with pseudoorbits indexed by positive integers and with one-step errors converging to zero in the future, usually called limit pseudo-orbits, and with orbits shadowing them in the limit (see Section 2 for precise definitions).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…It is called the two-sided limit shadowing property. The dynamics of systems with such property has been studied (see [5], [6], [7] and [10]) and the class of homeomorphisms satisfying it is growing (see [2], [8] and [9]). It is known that the two-sided limit shadowing property differs in several ways from the limit shadowing property.…”

Section: Introduction and Statement Of Resultsmentioning

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 similar procedure as in [6] follows to prove that f is a homeomorphism. Since for any δ > 0 there is K ∈ N + such that 1 k < δ for all k ≥ K, Γ 1 K (p) contains countably many points from E. Therefore, f is not pointwise expansive.…”

Section: Pointwise Measure Expansivitymentioning

confidence: 91%

Measure Expansivity and Specification for Pointwise Dynamics

Das

Khan

Das

2019

Bull Braz Math Soc, New Series

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We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy. (2010): 54H20, 37C50, 37B40 Mathematics Subject Classifications

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N-expansive homeomorphisms with the shadowing property

Cited by 46 publications

References 12 publications

Beyond topological hyperbolicity: The L-shadowing property

Beyond topological hyperbolicity: The L-shadowing property

Suspensions of homeomorphisms with the two-sided limit shadowing property

Measure Expansivity and Specification for Pointwise Dynamics

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