We prove that IHS A , the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is ℵ 0 -stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set.Similarly, APr A , the theory of atomless probability algebras equipped with a generic automorphism is ℵ 0 -stable up to perturbation.However, not allowing perturbation it is not even superstable.