Topological Models for Generators by MICHAEL SEARS* Department of Applied Mathematics University of the Witwatersrand Johannesburg, South Africa 1. Introduction. The striking similarity between the idea of a generator for a dynamical system and a generator for a topological cascade was pointed out by H. B. Keynes and J. R. Robertson in [2]. If(X, Z, S,/z) is a dynamical system, then a generator ~ is a finite partition of X such that the collection {SIP; P c ~ and i is an integer} generates Z. If (X, T) is a cascade consisting of a compact space and a homeomorphism T from X onto itself, then a generator is a finite open cover of X such that ~i=-o~T Ui is at most one point, where each Ui ~ °k'. But if q/is a generator, then {TiU; Uc ql and i is an integer} is a sub-basis for the topology on X, and in fact if X is first countable, this is a necessary and sufficient condition for q/to be a generator. The object of this paper is to show that to each generator for a dynamical system we can associate a cascade with a corresponding generator and to study the properties of this correspondence. Since a generator for a dynamical system does not depend on the measure itself but only on the or-algebra, we consider the structures discussed by I. Kluvfinek and B. Rie6an in [3]: triples (iV, Z, S) consisting of a set J(, a ~-algebra i~ of subsets of X and an invertible measurable transformation S from X onto itself (i.e., an automorphism of Z). (X, E, S) is called an algebra with automorphism. A generator ~ for (X, Z, S) is defined exactly as above and (X, E, S, ~) is then called an algebra with automorphism and generator. With the problem in these terms we investigate the idea of a topological model for (X, Y, S, ~). The obvious choice for a cascade would seem to be the Stone space corresponding to Z, but this "compactification" is far too large; in general, it is not metrizable and so cannot possess a generator. How- ever, we will see that smaller "compactifications" of Z can be constructed by using generators. 2. Existence and Uniqueness of Topological Models. We define precisely the relationship we require between (X, E, S, ~) and an associated cascade. * This research was supported by the Commonwealth Scientific and Industrial Research Organisation of Australia. 32 MATHEMATICAL SYSTEMS THEORYI Wol. 7. No. 1. © 1973 by Springer-Verlag New York Inc.