Semigroup Forum Vol. 43 (1991) 202-217 9 1991 Springer-Verlag New York Inc. RESEARCH ARTICLE Endomorphic actions of gN on the torus group W. A. F. Ruppert* Communicated by K, H. Hofmann Dedicated to Karl Prachar on the occasion of his 65 th birthday 1. Introduction Let X be a compact space and f:X ~ X a continuous selfmap of X. Then the iterates X --* X , x H fn(z) = fof ..... f(x), naturally induce an action of the discrete semigroup N on X, given by X x N ~ X,(x,n) H f"(x). In topological dynamics such actions are often studied by means of the so-called Ellis semigroup: the pointwise closure of the set {fn I n E N} in X x. The Ellis semigroup E(f) is a compact semigroup whose multiplication is in general continuous only on one side, containing as a dense central subset the set of all elements corresponding to the continuous maps fn. A basic result says that E(f) is a continuous homomorphic image of the semigroup /3N, so we also have a one-sidedly continuous action X x 3N --* X of/3N on X. From the viewpoint of harmonic analysis the case where X is the torus group I and f an endomorphism is of particular interest. Every continuous endomorphism of 9 is a power operator of the form P: x H x ~ , for some integer a; it induces the action -I- x NI --. T, (x,n) ~ x a" . The present paper is devoted to the study of this action for some fixed a > 2. The assumption a > 2 is not very stringent: for a = 0,1 we get the trivial actions where P is either the 0-endomorphism or the identity, and for a < 0 the action can be described easily in terms of the the inversion x ~ z -1 and the "positive action" induced by x ~ x -~ . The restriction a # 2 is motivated mainly by technical convenience, since to handle also the case a = 2 in many instances tedious extra technicalities would have been necessary. * The author gratefully acknowledges the financial support he received from the Alexan- der yon Humboldt Foundation during the prepapration of this paper.