The dynamical properties of even shift space

Abstract

A dynamical system is a system that evolves with time. Research in the field of dynamical system is largely focused on the nature of chaos on that system. Nowadays, there are various definitions of chaotic dynamical systems. However, the most well-known definition of chaos is Devaney chaos that states three chaotic conditions in its definition; sensitivity dependence on initial conditions, transitivity and density of periodic points. In this paper, we are investigating the presence of chaotic behavior in a discrete space, even shift. The even shift space is a space of all infinite sequences over symbols 0 and 1 such that between any two 1’s there are an even number of 0’s. By the end of this investigation, we prove that even shift space is not only Devaney chaos but also satisfies some other stronger chaotic conditions i.e. totally transitive, topologically mixing, blending, and locally everywhere onto.