New Conceptions of Transitivity and Minimal Mappings

Abstract

The concepts of topological δ- transitive maps, α-type transitive maps, δ-minimal and α-minimal mappings were introduced by M. Nokhas Murad Kaki. In this paper, the relationship between two different notions of transitive maps, namely topological δ-type transitive mapsandtopological α-type transitive maps has been studied and some of their properties in two topological spaces (X, τδ)and (X, τα), τδ denotes the δ–topology (resp. τα denotes the α–topology) of a given topological space (X, τ) has been investigated.. Also, we have proved that there exists a dense orbit in X, where X is locally compact Hausdorff space and τ has a countable basis. The main results are the following propositions:Every topologically α-type transitive map is a topologically transitive map which implies topologically δ- transitive map, but the converse not necessarily true., and every α-minimal map is a minimal map which implies δ- minimal map in topological spaces, but the converse not necessarily true. Finally, we have proved that a map which is γr- conjugated to γ-transitive (resp. γ-minimal, γ-mixing) map is γ-transitive (resp. γ-minimal, γ-mixing).