e*−Transitivity as an Extension to Chaos Notions

Abstract

The introduction of e^∗−transitivity is motivated by the broader concept of e^∗−open sets, representing a generalised extension of various open set concepts. In this paper, we prove that the e^∗−transitivity implies the classical transitivity and many other types of transitivity such as b−transitivity, δ−transitivity, α−transitivity, and sr−transitivity. Therefore, exploring the properties of e^∗−transitivity holds significance, as these properties extend to other forms of transitivity. Additionally, we introduce several e∗−chaos notions that serve as analogoues, to their classical counterparts. These notions include total e^∗−transitivity, e^∗−mixing, e^∗−locally everywhere onto (abbreviated as e^∗−l.e.o.), and weak e^∗−blending. Subsequently, we demonstrate the interrelations among these newly introduced e^∗−chaos notions and illustrate their connections to classical chaos concepts. This comparative analysis not only enriches our understanding of e^∗−transitivity but also introduce new notions of chaos based on e^∗−transitivity.