Dynamical compactness and sensitivity

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 syste...

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Bibliographic Details
Published inJournal of Differential Equations Vol. 260; no. 9; pp. 6800 - 6827
Main Authors Huang, Wen, Khilko, Danylo, Kolyada, Sergiĭ, Zhang, Guohua
Format Journal Article
LanguageEnglish
Published Elsevier Inc 05.05.2016
Subjects
Dynamical topology
Li–Yorke sensitivity
Lyapunov numbers
Multi-sensitivity
Topological weak mixing
Transitive compactness
secondary
Dynamical topology
Multi-sensitivity
Lyapunov numbers
Transitive compactness
Li–Yorke sensitivity
Topological weak mixing
primary
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Summary: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 between 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.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2016.01.011