Sensitivity of Iterated Function Systems Under the Product Operation

Waseem, Anwar; Lu, Tianxiu; Yang, Xiaofang

doi:10.1007/s00025-022-01669-6

Cited by 5 publications

(1 citation statement)

References 26 publications

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“…Sensitive dependence on initial conditions, introduced by Guckenheimer in [1], is widely recognized as a fundamental concept in chaos theory [2][3][4][5]. However, it should be noted that sensitivity is a topological concept rather than a measurable one.…”

Section: Introductionmentioning

confidence: 99%

Measurable Sensitivity for Semi-Flows

Quan,

Lu,

et al. 2023

Mathematics

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Sensitive dependence on initial conditions is a crucial characteristic of chaos. The concept of measurable sensitivity (MS) was introduced as a measure-theoretic version of sensitive dependence on initial conditions. Their research demonstrated that MS arises from light mixing, indicates a finite number of eigenvalues for a transformation, and is not present in the case of infinite measure preservation. Unlike the traditional understanding of sensitivity, MS carries up to account for isomorphism in the sense of measure theory, which ignores the function’s behavior on null sets and eliminates dependence on the chosen metric. Inspired by the results of James on MS, this paper generalizes some of the concepts (including MS) that they used in their study of MS for conformal transformations to semi-flows, and generalizes their main results in this regard to semi-flows.

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Section: Introductionmentioning

confidence: 99%

Measurable Sensitivity for Semi-Flows

Quan,

Lu,

et al. 2023

Mathematics

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Further Studies of Topological Transitivity in Non-Autonomous Discrete Dynamical Systems

Pi,

Lu,

Anwar

et al. 2024

jaac

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Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems

Zhou,

Lu,

Zhao

2023

Axioms

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This paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) (H,f1,∞) and its induced set-valued discrete dynamical systems (K(H),f¯1,∞). Specifically, it explores the chaotic properties of these systems. The main finding is that f1,∞ is Devaney chaotic if and only if f¯1,∞ is Devaney chaotic in we-topology. The paper also provides similar conclusions for weak mixing, mixing, mild mixing, chain-transitivity, and chain-mixing in non-autonomous set-valued discrete dynamical systems (NSDDSs). Additionally, the paper proves that weak mixing implies sensitivity in NSDDSs.

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Sensitivity of Iterated Function Systems Under the Product Operation

Cited by 5 publications

References 26 publications

Measurable Sensitivity for Semi-Flows

Measurable Sensitivity for Semi-Flows

Further Studies of Topological Transitivity in Non-Autonomous Discrete Dynamical Systems

Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems

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