A zero-one law for dynamical properties

For any countable group satisfying the \weak Rohlin property", and for each dynamical property, the set of-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice Z d ; indeed, all countable discrete amenable groups. For an arbitrary countable group, let A be the set of-actions on the unit circle Y. We establish an Equivalence theorem by showing that a dynamical property is Baire/meager/residual in A if and only if it is Baire/meager/residual in the set of shift-invariant measures on the product space Y. x1 Introduction Halmos's book Ergodic Theory introduced many of us to the study of determining which dynamical properties are generic (i.e, topologically residual) in the so-called \coarse topology" on transformations. For instance, \weak-mixing" is generic, whereas \mixing" is not, Hal, pp. 77,78]. The exploration of this notion of genericity became an active research area; see CP] and CN] for results and extensive bibliographies. It has often been the case, when a property has failed to be generic, that further investigation has shown its negation to be generic. This suggests that there is a type of \zero-one" law operating for dynamical properties {each is either meager or residual; that is, either a rst-category set (a countable union of nowhere-dense sets) or the complement of such. In 1993, we found a demonstration of this for certain acting groups , for dynamical properties which are Baire-measurable. We rst present the brief proof in the abstract framework of a group acting as homeomorphisms of a topological space A , where A fulllls the Baire Category Theorem in that each residual set is dense. We call such a space a BaireCat space, and say that a subset of a topological space is BaireCat if it is a BaireCat space in its induced topology. y We then show how this abstract framework applies when A is the space of measure-preserving-actions with playing the role of its group of isomorphisms. In this instance, A will be a Polish space, that is, homeomorphic to a complete separable metric space. y A \Baire space" is the traditional name for what we call a BaireCat space. The traditional meanings of \Baire set" and \Baire space" are not related by the induced topology. In view of this, we have revised terminology for this article.