Dense periodicity property and Devaney chaos on shifts spaces

Abstract

Transitivity and dense periodic points are two ingredients of Devaney chaos. Locally everywhere onto, totally transitivity and mixing are other chaos notions which are stronger than transitivity and have been studied widely. In this paper, we will look at other recently introduced chaos notion, which is stronger than dense periodic points. This paper will examine the role of the strong dense periodicity on some shifts spaces.

Some Chaotic Results of Product on Zero Dimension Spaces

“…Then , there exists that some of them are singletons, empty set or the whole * + , which have trivial dynamics and are not of our interest. There exists six distinct one-step shift of finite types over two symbols, and , with sets of forbidden blocks * + * + * + * + * + and * + respectively [1]. They are shown below through matrices and their own graph…”

“…Several efforts have been made to give the notion of chaos a mathematically precise meaning. However, chaos is not simple to define and has no universally concordant definition [1].…”

“…is topologically transitive, the periodic points of is dense, and posses sensitive dependence on initial conditions or simply (SDIC)) [1,2].…”

The Chaotic Properties of Increasing Gap Shifts

“…It turns out, however, that strengthening this dense periodicity property yields different results. On shift of finite type, the implication is true [13] but not the case on the unit interval [10]. The study of chaotic behavior on shift of finite type has been done extensively in various approaches.…”