Relationship between New Types of Transitive Maps and Minimal Systems.

The concepts of topological ?-transitive maps, ?-transitive maps, ?-minimal and ?-minimal mappings were introduced by M. Nokhas Murad. In this paper, I study the relationship between two different notions of transitive maps, namely topological ?- transitive maps, topological ?-transitive maps and investigate some of their properties in two topological spaces (X, ??) and (X, ??), ?? denotes the ?–topology (resp. ?? denotes the ?–topology) of a given topological space (X, ? ).. The two notions are defined by using the concepts of ?-irresolute map and ?-irresolute map respectively Also, we study the relationship between two types of minimal mappings, namely, ?-minimal mapping and ?-minimal mapping, and I will prove that the properties of ?-transitive, ?-mixing and ?-minimal maps are preserved under ?r-conjugacy The main results are the following propositions: 1) Every topologically ?-transitive map is a topologically transitive map which implies topologically ?- transitive map, but the converse not necessarily true. 2) Every ?-minimal map is a minimal map which implies ?- minimal map in topological spaces, but the converse not necessarily true.