Multi-sensitivity with respect to a vector for semiflows
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Multi‑sensitivity with respect to a vector for semiflows Rahul Thakur1 · Ruchi Das1 Received: 28 September 2019 / Accepted: 5 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The notions of multi-sensitivity with respect to a vector, N -sensitivity and strong multi-sensitivity are introduced and studied on semiflows under the action of the most general possible semigroups. Using the concept of Furstenberg families, some other stronger forms of sensitivity are also studied. Moreover, all these notions are studied on hyperspaces and on the product of semiflows. Keywords Semiflow · Sensitivity · Multi-sensitivity · Hyperspace
1 Introduction In the past decade, significant study of dynamical systems under general semigroup actions has been done. Throughout the paper, by a semiflow we mean an action of a topological semigroup on a metric space. After the introduction of mathematical concept of chaos by Li and Yorke [7], the theory of chaos has seen major developments and it is still one of the important phenomena in the theory of semiflows, which is being widely investigated. Sensitive dependence on initial conditions, or sensitivity, is an important ingredient of chaos theory as it characterizes the unpredictability of a chaotic phenomenon. This notion was introduced by Auslander and Yorke [1], and later it was popularized by Devaney [3]. For measuring the strength of sensitivity, Moothathu [11] initiated a preliminary study of stronger forms of sensitivity. The author has introduced and studied three stronger forms of sensitivity, namely cofinite sensitivity, multi-sensitivity and syndetic sensitivity. In [13], Tan and Zhang have introduced a more general description of sensitivity via Furstenberg
Communicated by Jimmie D. Lawson. * Ruchi Das [email protected] Rahul Thakur [email protected] 1
Department of Mathematics, University of Delhi, Delhi 110007, India
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families. On semiflows, several other stronger forms of sensitivity have been introduced and studied by many researchers [8, 17]. Using the vectors of ℕr , Chen and others [2] have introduced and studied stronger notions of transitivity, such as multi-transitivity with respect to a vector, multitransitivity and strong multi-transitivity for discrete systems (by a discrete system we simply mean an action of the non-negative integers on a metric space). Later these concepts have been generalized for general semiflows by Zeng [19]. Recently, Jiao and others [5] have introduced and studied several stronger forms of sensitivity, such as multi-sensitivity with respect to a vector, N -sensitivity and strong multisensitivity for discrete systems. Therefore, a natural question that arises is whether these notions can be introduced for general semiflows. The relationships between the various forms of sensitivity of a discrete system and its induced hyperspace (when phase space is a hyperspace) are studied by Sharma and Nagar [12]. Wu and others [18] have studied sensitivity o
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