Abstract We study the cyclic, supercyclic and hypercyclic properties of a composition operator C ϕ on Segal-Bargmann space ℋ(ℰ), where ( z ) = Az + b , A is bounded linear ℰ, ∈ ℰ with || ⩽ 1 * belongs to range I – )½. Specifically, under some conditions symbol we show that if cyclic then A* but converse need not be true. also cyclic. Further there no ℋ(ℰ) for certain class symbols .