Recent development of chaos theory in topological dynamics

Li, Jian; Ye, Xiangdong

doi:10.1007/s10114-015-4574-0

Cited by 95 publications

(54 citation statements)

References 107 publications

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“…First of all, the proximal pair can be characterized via ultrafilter as follows. Another customary description of chaos is sensitivity to initial conditions (cf., e.g., [11,20] for Z + -systems and [7,8] for R + -systems):…”

Section: Li-yorke Chaotic Pairs and Sensitivity (I)mentioning

confidence: 99%

Chaotic dynamics of minimal center of attraction of discrete amenable group actions

Chen

Dai

2017

Journal of Mathematical Analysis and Applications

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Section: Li-yorke Chaotic Pairs and Sensitivity (I)mentioning

confidence: 99%

Chaotic dynamics of minimal center of attraction of discrete amenable group actions

Chen

Dai

2017

Journal of Mathematical Analysis and Applications

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“…We thank MPIM for providing an ideal setting. We also thank Xiangdong Ye for sharing his joint work [31,41], and thank the referee for comments that have resulted in substantial improvements to this paper.…”

Section: It Is Not Striking That These New Lyapunov Numbersmentioning

confidence: 99%

“…Furstenberg started a systematic study of transitive dynamical systems in his paper on disjointness in topological dynamics and ergodic theory [11], and the theory was further developed in [13] and [12]. The main motivation for this paper comes from [39], [2], [24], [5], [14], [22], [38], [37] and recent papers [29], [34] and [31], which discusses a dynamical property called transitive compactness examined firstly for weakly mixing systems in [5]: transitive compactness is quite related to but different from the property of transitivity, it will be equivalent to weak mixing under some weak conditions, and it presents some kind of sensitivity of the system.…”

Section: Introductionmentioning

confidence: 99%

“…[16], [3], [15], [2], [4], [5], [1], [21], [30]). For the recent development of sensitivity in topological dynamics see for example the survey [31] by Li and Ye. According to the works by Guckenheimer [17], Auslander and Yorke [7] a dynamical system (X, T ) is sensitive if there exists δ > 0 such that for every x ∈ X and every neighborhood U x of x, there exist y ∈ U x and n ∈ N with d(T n x, T n y) > δ. Such a δ is also called a sensitive constant of the system (X, T ).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamical compactness and sensitivity

Huang¹,

Khilko²,

Kolyada³

et al. 2016

Journal of Differential Equations

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Abstract. To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system (X, T ) given by a compact metric space X and a continuous surjective self-map T : X → X. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is not proximal, and that any multi-sensitive system has positive topological sequence entropy. Moreover, we show that multi-sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for M-systems. We also give a quantitative analysis for multi-sensitivity of a dynamical system.

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“…Instead of constructing uncountable scrambled sets directly, Kuratowski-Mycielski theorem has been applied extensively to show the existence of "large" scrambled sets in topological dynamics. We refer the reader to [6] and [17] for recent advances on this topic.…”

Section: Introductionmentioning

confidence: 99%