Pointwise Dynamics Under Orbital Convergence
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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 \(\alpha \)-persistence at a point need not be preserved under pointwise convergence.
KeywordsExpansivity Shadowing Transitivity Topological Stability Chaos
Mathematics Subject ClassificationPrimary 54H20 Secondary 40A30
The first author is supported by CSIR-Junior Research Fellowship (File No.-09/045(1558)/ 2018-EMR-I) of Government of India. The authors express sincere thanks to the reviewer for suggestions.
- Walters, P.: On the pseudo orbit tracing property and its relationship to stability. In the structure of attractors in dynamical systems, pp. 231–244. Springer, Berlin, Heidelberg (1978)Google Scholar