Two New Types of Chaotic Maps and Minimal Systems

Abstract

In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.