One of the roots of the modern study of  hypercyclicity comes from an intriguing observation of  G .D. Birkhoff”s concerning the orbits of translation operators acting on the space of entire functions.This article contains 4 chapter,that in first chapter we state the elementary difinations and proof some necesury theorem. And in late we state the history of hypercyclic operators.In section 1 of chapter 2 we cheking the hypercyclic and supercyclic vectors and some properties from those vectors. In continuation we introduce the invariant subspace, afterward, we state hypercyclic criterion. In continuance, we state hypercyclic criterion theorem and proof them.In this chapter we also proof that, If all non-zero vectors of Hilbert space are hypercyclic for operator T , Then T have no non-trivial closed invariant subsets. In section2 of this chapter we deliberation Invariant manifolds of hypercyclic vectors for the real scalar case. In chapter 3 we deliberation the spaces that admit hypercyclic operators whit hypercyclic adjoints. Also, we show that each hypercyclic operator on rael locally convex vector space, have a dense variant linear manifold of hypercyclic vectors. For this,Whit state the Salas theory,  we introduce such spaces.Whit definition of showther basis , orthogonal basis and shrinking basis, we introduce some of the spaces that admit hypercyclic operator whit hypercyclic adjoint.In chapter 4 we study hypercyclic differentiation operators, For this, we deliberation the space of entire functions of one complex variable, endowed with the topology of uniform convergence on compact subsets of the plane.

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