Relationship between New Types of Transitive Maps and Minimal Systems
|Research Area:||Volume 4 Issue 6, Nov. 2013||Year:||2013|
|Type of Publication:||Article||Keywords:||Topologically Transitive, Irresolute, Transitive, Dense|
The concepts of topological α-transitive maps, θ-transitive maps, α-minimal and θ-minimal mappings were introduced by M. Nokhas Murad. In this paper, I study the relationship between two different notions of transitive maps, namely topological α- transitive maps, topological θ-transitive 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-irresolute map and θ-irresolute map respectively Also, we study the relationship between two types of minimal mappings, namely, α-minimal mapping and θ-minimal mapping, and I will prove that the properties of θ-transitive, θ-mixing and θ-minimal maps are preserved under θr-conjugacy The main results are the following propositions: 1) Every topologically α-transitive map is a topologically transitive map which implies topologically θ- transitive 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.
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