Throughout this paper (X, d) will be a metric space with metric d, and h a homeomorphism of X onto itself. For any real number r > 0, and p ∊ X, U(p, r) will denote the open r - sphere about p. Any point p ∊ X is called regular [3] if for any given ∊ > 0 there exists a δ > 0 such that d(p, y) <δ implies d(hn(p), hn(y)) < ∊ for all integers n, where hn denotes the iterates of h for n > 0, of h-1 for n < 0, and h0 is the identity. Any point of X which is not a regular point i s called an irregular point. Let I(h) denote the set of all the irregular points of X and R(h) = X-I(h). Lim inf and Lim sup are defined as in [4].