Conceptions of Transitive Maps in Topological Spaces.

The concepts of topological ?-type transitive maps, ?-transitive maps, ?-minimal and ?-minimal mappings were introduced by Mohammed Nokhas Murad. . In this paper, the relationship between two different notions of transitive maps, namely topological ?-type transitive maps and topological ?-type transitive maps has been studied and some of their properties in two topological spaces (X, ??) and (X, ??), ?? denotes the ? –topology(resp. ?? denotes the ?–topology) of a given topological space (X, ? ) has been investigated.. Also, we have proved that there exists ?- dense orbit in X, where X is locally ?- compact Hausdorff space and ?? has a countable basis Also, I study the relationship between two types of minimal mappings, namely, ?-minimal mapping and ?-minimal mapping, and I will prove that the properties of ?-type transitive, ?-mixing and ?-minimal maps are preserved under ?r-conjugacy.