Different Types of Maps Route to Chaos in Bi-infinite Symbol Space with Strong Chaotic Features

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.