International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-12, December 2016 51 www.ijeas.org Abstract—Some results of strongly continuous semigroup ( -semigroup) of functional analysis of bounded linear operator defining on a separable Banach spaces like hypercyclic, topologically transitive, chaotic, mixing, weekly mixing and topologically ergrodic have been discussed. The generalization of hypercyclic, topologically transitive, chaotic, mixing, weakly mixing and topologically ergrodic for the direct sum of two and /or hence to n -semigroup strongly continuous dynamic system of semigroups in infinite dimensional separable Banach space have been developed with proofs and discusses Index Terms—semigroup, ergrodic, dynamic system. I. INTRODUCTION Let be a separable infinite dimensional Banach space. A one-parameter family of continuous (bounded) linear operators on X is a strongly continuous semigroup ( -semigroups) if , and , for every . A -semigroup is said to be topologically transitive if for any nonempty open subsets of X, there exists some , such that [1], and it is said to be mixing if there exists some such that the condition hold for all [2]. On the other hand a -semigroup is said to be hypercyclic if there exists some whose orb is dense in X in this case we say that is called hypercyclic vector for this semigroup [2]. Note that the hypercyclicity and transitivity for a -semigroup are equivalent on on a separable Banach space [3]. It's clear that if is a hypercyclic operator for some , then the semigroup is hypercylic. On the other hand if the semigroup is hypercylic then is hypercylic operator for all [1]. Also a -semigroup is said to be weakly mixing if is Radhi.Ali Zaboon, Department of mathematics, College of Science, Al- Mustansiriyah, University, Baghdad, IRAQ Arwa Nazar Mustafa, Department of mathematics, College of Science, Al- Mustansiriyah, University, Baghdad, IRAQ Rafah Alaa Abdulrazzaq, Department of mathematics, College of Science, Al- Mustansiriyah, University, Baghdad, IRAQ topologically transitive [1], and it's said to be a topologically ergodic if for every non-empty open subsets of X, the set is syndetic, that is does not contains arbitrarily long intervals[2]. A point is called a periodic point of if there is some such that [3]. A -semigroup on X is called chaotic if it is hypercyclic and its set of periodic points is dense in X. Note that if are separable Banach spaces, then the space is a Separable Banach space, and if and are -semigroups on X and Y respectively, then the direct sum of and is a -semigroup on defined by [2]. II. SOME PROPERTIES THAT PRESERVED UNDER QUASICONJUAGACY: Definition(2.1) [2]: Let and be -semigroups on and Y, respectively then is called quasiconjugate to if there exists a continuous map with dense range such that . If can be chosen to be a homeomorphism then and are called conjugate. Definition(2.2) [2]: A property P of -semigroup is said to be preserved under (quasi) conjugacy if any -semigroup that is (quasi)conjugate to a -semigroup with property P also possesses property P. Results On Strongly Continuous Semigroup Radhi.Ali Zaboon, Arwa Nazar Mustafa, Rafah Alaa Abdulrazzaq