Let f n ( x ) be the n t h iterate of a function in some interval [ 0 , c ] . It is known that if f ( x ) ∼ x − x α , 1$"> α > 1 , then f n ( x ) ∼ A n a for some A and a . In this paper we prove a converse of this theorem: The rate of convergence of the iterates determines the form of a function.