Diffeomorphisms with the<i>s</i>-limit shadowing property

Sakai, Kazuhiro

doi:10.1080/14689367.2012.691960

Cited by 9 publications

(5 citation statements)

References 11 publications

(5 reference statements)

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“…Meanwhile, shadowing property is also generalized to various other forms. For example, there are studies on limit-shadowing [75], s-limit-shadowing [61,83,77], average-shadowing [56], asymptotic-average-shadowing [31], thick shadowing [23], d-shadowing [27,65], and ergodic shadowing [33].…”

Section: Pseudo-orbit Tracing Propertiesmentioning

confidence: 99%

Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center

Dong¹,

Tian²

2018

Preprint

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In [30] different statistical behavior of dynamical orbits without syndetic center are considered. In present paper we continue this project and consider different statistical behavior of dynamical orbits with nonempty syndetic center: Two of sixteen cases appear (for which other fourteen cases are still unknown) in transive topologically expanding or hyperbolic systems and are discovered to have full topological entropy for which it is also true if combined with non-recurrence and multifractal analysis such as quasi-regular set, irregular set and level sets. In this process a strong entropy-dense property, called minimal-entropy-dense, is established.In particular, we show that points that are minimal (or called almost periodic), a classical and important concept in the study of dynamical systems, form a set with full topological entropy if the dynamical system satisfies shadowing or almost specification property.

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Section: Pseudo-orbit Tracing Propertiesmentioning

confidence: 99%

Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center

Dong¹,

Tian²

2018

Preprint

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“…Meanwhile, shadowing property is also generalized to various other forms. For example, there are studies on limit-shadowing [81], s-limit-shadowing [64,90], average-shadowing [60], asymptotic-average-shadowing [31], thick shadowing [22], dshadowing [28,68], and ergodic shadowing [34].…”

Section: Pseudo-orbit Tracing Propertiesmentioning

confidence: 99%

Twelve Different Statistical Future of Dynamical Orbits: Empty Syndetic Center

Dong¹,

Hou²,

Tian³

2017

Preprint

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In the theory of dynamical systems, a fundamental problem is to study the asymptotic behavior of dynamical orbits. Lots of different asymptotic behavior have been learned including different periodic-like recurrence such periodic and almost periodic, the level sets and irregular sets of Birkhoff ergodic avearge, Lyapunov expoents. In present article we use upper and lower natural density, upper and lower Banach density to differ statistical future of dynamical orbits and establish several statistical concepts on limit sets, in particular such that not only different recurrence are classifiable but also different non-recurrence are classifiable. In present paper we mainly deal with dynamical orbits with empty syndetic center and show that twelve different statistical structure over expanding or hyperbolic dynamical systems all have dynamical complexity as strong as the dynamical system itself in the sense of topological entropy. Moreover, multifractal analysis on various non-recurrence and Birkhoff ergodic averages are considered together to illustrate that the non-recurrent set has rich and colorful asymptotic behavior from the statistical perspective, although the non-recurrent set has zero measure for any invariant measure from the probabilistic perspective.Roughly speaking, on one hand our results describe a world in which there are twelve different predictable order in strongly chaotic systems but also there are strong chaos in any fixed predictable order from the viewpoint of dynamical complexity on full topological entropy; and on other hand we find that various asymptotic behavior such as (non-)recurrence and (ir)regularity from differnt perspecitve survive togother and display strong dynamical complexity in the sense of full topological entropy. In this process we obtain two powerful ergodic properties on entropy-dense property and saturated property.

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“…In [26,27], various limit shadowing properties are examined in relation to the notion of hyperbolicity and stability. The set of Ω-stable diffeomorphisms of a smooth closed manifold is characterized as the C 1 -interior of the set of diffeomorphisms satisfying the limit shadowing property [26].…”

Section: Introductionmentioning

confidence: 99%

“…Shadowing has been the subject of much interest in the qualitative study of dynamical systems [3,25], and various shadowing properties have been defined in the course of such studies so far. The limit shadowing property introduced in [11] is one of the variants of the shadowing property, which focuses on the possibility of asymptotic shadowing of pseudo orbits whose one-step errors are converging to zero, and it is a subject of ongoing research, see [5,8,9,14,15,18,21,23,26,27].…”

Section: Introductionmentioning

confidence: 99%

On the shadowing and limit shadowing properties

Kawaguchi

2020

Fund. Math.

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We study the relation between the shadowing property and the limit shadowing property. We prove that if a continuous self-map f of a compact metric space has the limit shadowing property, then the restriction of f to the non-wandering set satisfies the shadowing property. As an application, we prove the equivalence of the two shadowing properties for equicontinuous maps.2010 Mathematics Subject Classification. 37C50; 54H20; 37B20.

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Diffeomorphisms with thes-limit shadowing property

Cited by 9 publications

References 11 publications

Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center

Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center

Twelve Different Statistical Future of Dynamical Orbits: Empty Syndetic Center

On the shadowing and limit shadowing properties

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