We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cφ on the Segal-Bargmann space H(ϵ), where φ(z) - Az + b, A is a bounded linear operator on ϵ, b ∈ ϵ with |A| ≤ 1 and A∗b belongs to the range of (I - A∗A)1/2. Specifically, under some conditions on the symbol φ we show that if Cφ is cyclic then A∗ is cyclic but the converse need not be true. We also show that if Cφ∗ is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space H(ϵ) for certain class of symbols φ. © 2022 G. Ramesh et al., published by De Gruyter.