ISRAEL JOURNAL OF MATHEMATICS, Vol. 34, Nos. 1-2, 1979 ALMOST TOPOLOGICAL DYNAMICAL SYSTEMS BY MANFRED DENKER AND MICHAEL KEANE ABSTRACT For the notion of finitary isomorphism, which arises in many examples in ergodic theory, we prove some basic theorems about invariants, representations and the central limit theorem in shift spaces. Flows arising in physics or ergodic theory often present singular behavior on a negligible set of states, although their topological properties are important. Almost topological dynamical systems form a suitable model for describing, both topologically and metrically, asymptotic comportment of these flows. They are defined as measure-preserving transformations acting on a compact metric space which are continuous after removal of a negligible set of singularities. Much attention in recent years has been paid in ergodic theory to the interplay of topological and metric properties (see [10], [18]), and we attempt here to present the foundations for a systematic study. The first section contains the basic definitions of almost topological dynamical systems and finitary homomorphisms and isomorphisms of such systems. De- scriptions of invariants for finitary isomorphisms are contained in the second paragraph, the most important of which is strict ergodicity (Theorem 7). From this it follows that metrical isomorphism classes are in general larger than finitary classes, in particular in the case of Bernoulli schemes. In the third section we show that any aperiodic almost topological dynamical system is finitarily isomorphic to a shift dynamical system, and in the ergodic case with finite entropy the existence of a finite (almost topological) generator with the minimal number of symbols is derived. The last section deals with finitary homomorphisms between two shift dynamical systems. Such homomorphisms can be effected by a sequential coding procedure, and we look at those homomorphisms for which the coding has a finite expectation (Definition 22). In this case, we show that if the coordinate process of a dynamical system satisfies a certain mixing condition, then any Received March 16, 1978 139