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DOI:http://dx.doi.org/10.26855/jamc.2020.09.006

Chaotic Features of the Forward Shift Map on the Generalized m-Symbol Space

Hena Rani Biswas1,*, Md. Shahidul Islam2

1Department of Mathematics, University of Barishal, Barishal-8200, Bangladesh.

2Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh.

*Corresponding author: Hena Rani Biswas

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Date: September 30,2020 Hits: 264, How to cite this paper

Abstract

In this paper, we study some chaos related properties of the forward shift map on the generalized one-sided symbol space . We consider several notions of chaos available in the contemporary literature. In this paper, we prove that  is Devaney chaotic, Auslander-Yorke’s chaotic and generically δ-chaotic. We prove that  is exact Devaney chaotic and as a consequence is mixing Devaney Chaotic and weak mixing Devaney Chaotic. We also provide examples to show that the forward shift map  on  is topologically conjugate to the map on the space or equivalently conjugate to the map fm (x)=mx(mod 1) on the space R/Z. Finally, we give a counterexample to prove that not all topologically transitive maps are totally transitive. We also give an example of a continuous function that is topologically transitive but not chaotic.

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Chaotic Features of the Forward Shift Map on the Generalized  m-Symbol Space
How to cite this paper: M. G. Sobamowo. (2020) Chaotic Features of the Forward Shift Map on the Generalized m-Symbol Space. Journal of Applied Mathematics and Computation, 4(3), 104-112.
DOI: http://dx.doi.org/10.26855/jamc.2020.09.006

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