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Journal of Vibration Testing and System Dynamics 9(1) (2025) 97--104 | DOI:10.5890/JVTSD.2025.03.007

Hena Rani Biswas, Ruma Rani, Sakibul Islam, Umme Habiba Khatun

Department of Mathematics, University of Barishal, Barishal-8254, Bangladesh

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Abstract

The main purpose of this study is to introduce a generalization of the shift two-sided map, that is, the generalized two-sided shift map. In this paper, we discuss periodic points that are dense in $\Sigma_{m}$ regarding the generalized shift map $\sigma_{n}$: ${\Sigma}_{m}\to{\Sigma}_{m}$. We consider a particular property of the dynamical system namely, the topologically transitivity. Here, we prove that the generalized shift map is topologically mixing and hence is topologically transitive on ${\Sigma}_{m}$. Few strong chaotic properties of the generalized shift map have been discussed. We know that the two-sided shift maps are automorphisms and the one-sided shift maps are endomorphisms. These maps can be conjugate or semi-conjugate to some automorphisms or endomorphisms which admit appropriate Markov partitions. We also discuss other maps such as $\alpha $-map and $\beta $-map are chaotic in bi-infinite symbol space.

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