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arXiv · 1509.08813

Dynamical compactness and sensitivity

Wen Huang·Danylo Khilko·Sergii Kolyada·Guohua Zhang
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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 $T:X \to X$. 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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Wen Huang, Danylo Khilko, Sergii Kolyada, Guohua Zhang. 2015-09-29. Dynamical compactness and sensitivity. https://arxiv.org/abs/1509.08813

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