Abstract

The concepts of topological #x3B3;-transitive maps, #x3B1;-transitive maps, #x3B3;-type chaotic and #x3B1;-type chaotic maps were introduced by M. Nokhas Murad Kaki. In this paper, I study the relationship between two different notions of transitive maps, namely topological #x3B1;- transitive maps, topological #x3B3;-transitive maps and investigate some of their properties in two topological spaces (X, #x3C4;#x3B1;) and (X, #x3C4;#x3B3;), #x3C4;#x3B1; denotes the #x3B1;#x2013;topology(resp. #x3C4;#x3B3; denotes the #x3B3;#x2013;topology) of a given topological space (X, #x3C4; ).. The two notions are defined by using the concepts of #x3B1;-irresolute map and #x3B3;-irresolute map respectively Also, we study the relationship between two new types of chaotic maps, namely, #x3B1;-type chaotic maps and #x3B3;-chaotic maps, and I will prove that the properties of #x3B1;- transitive, #x3B1;-type chaotic are preserved under #x3B1;r-conjugacy and #x3B3;-transitive, #x3B3;-chaotic maps are preserved under #x3B3;r-conjugacy The main results are the following propositions: 1) Every topologically #x3B3;-type transitive map is a topologically #x3B1;-type transitive map which implies topologically transitive map, but the converse not necessarily true.. 2) Every #x3B3;-type chaotic map is #x3B1;-chaotic map which implies chaotic map in topological spaces, but the converse not necessarily true..