Various Shadowing in Linear Dynamical Systems

Wu, Xinxing; Zhang, Xu; Ma, Xin

doi:10.1142/s0218127419500421

Cited by 13 publications

(4 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In a manner reminiscent of Li-Yorke's chaos, an equivalent characterization of Kato's chaos for a continuous map on a compact metric space was provided. For other types of chaos, refer to [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Kato Chaos in Linear Dynamics

Jiao

Wang

2023

Mathematics

View full text Add to dashboard Cite

This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics.

show abstract

Section: Introductionmentioning

confidence: 99%

Kato Chaos in Linear Dynamics

Jiao

Wang

2023

Mathematics

View full text Add to dashboard Cite

show abstract

“…A dynamical system has the shadowing property if every sufficiently precise trajectory is closed to some exact trajectory. The shadowing property has been developed intensively in recent years, and many authors obtained results about chaos and stability by studying the various type of shadowing (see [1,11,17,19,20,22,24,25,[27][28][29]). Wu et al [25] introduced the notion of M α -shadowing and proved that a dynamical system has the average shadowing property if and only if it has the M α -shadowing property for any α ∈ [0, 1).…”

Section: Introductionmentioning

confidence: 99%

Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

Wu¹

2021

Hacettepe Journal of Mathematics and Statistics

Self Cite

View full text Add to dashboard Cite

We introduce various new type of recurrent sets for finitely generated semigroups on noncompact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.

show abstract

“…A really convincing theory of topological dynamics exists only with the assumption that the phase space X, in addition to being metric, is also compact. Most general results concerning chaotic properties, like positive entropy, are obtained under this hypothesis [9,10,16,17,19]. Sometimes one prefers to consider a dynamical system on a non-metrizable topological space.…”

Section: Introductionmentioning

confidence: 99%