Pointwise Dynamics Under Orbital Convergence
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Pointwise Dynamics Under Orbital Convergence Abdul Gaffar Khan1 · Pramod Kumar Das2 · Tarun Das1 Received: 16 June 2019 / Accepted: 4 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform convergence and properties like topological stability and α-persistence at a point need not be preserved under pointwise convergence. Keywords Expansivity · Shadowing · Transitivity · Topological Stability · Chaos Mathematics Subject Classification Primary 54H20 ; Secondary 40A30
1 Introduction The idea of studying the behaviour of a dynamical system from pointwise viewpoint was initiated by Reddy. In the process of answering a question posed by Gottschalk to him, he introduced and studied pointwise expansivity, a strictly weaker notion than expansivity (Reddy 1970). The power and the beauty of pointwise dynamics got highlighted in the recent works including (Akin 1996; Morales 2016; Ye and Zhang
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Tarun Das [email protected] Abdul Gaffar Khan [email protected] Pramod Kumar Das [email protected]
1
Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India
2
School of Mathematical Sciences, Narsee Monjee Institute of Management Studies, Vile Parle, Mumbai 400056, India
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A. G. Khan et al.
2007). In Akin (1996), introduced the concept of chain continuity at a point which is a stronger version of shadowable point introduced in Morales (2016) by Morales and proved that every chain transitive continuous map with chain continuity at a point must be equicontinuous (Akin 1996, Corollary 2.3) which is interestingly not true for chain transitive systems with shadowable points. A decade later, authors have introduced (Ye and Zhang 2007) the concept of entropy point which worked as a key ingredient in the proof of (Moothathu 2011, Theorem 3). In this theorem, Moothatu has proved that certain kind of continuous map with shadowing property has positive entropy. Recently, Morales (2016) has proved that unlike expansivity, a homeomorphism on a compact metric space has shadowing if and only if each point is shadowable. In Kawaguchi (2017a), the notion of entropy point is used by Kawaguchi to show that the existence of certain kind of e-shadowable points implies positive entropy (Kawaguchi 2017b). In Das et al. (2019), authors have studied the relation of specification points with Devaney chaotic points and positive entropy of the system. In the same paper, authors have provided an example of
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