Quasi-disjointness in ergodic theory

Abstract

We define and study a relationship, quasi-disjointness, between ergodic processes. A process is a measure-preserving transformation of a measure space onto itself, and ergodicity means that the space cannot be written as a disjoint union of invariant pieces, unless one of the pieces is of zero measure. We restrict our attention to spaces of total measure one which also satisfy additional regularity properties. In particular, the associated Hilbert space of square-summable functions is separable. A simple class of examples is given by translation by a fixed element on a compact Abelian metrizable group, such processes being known as Kronecker processes. We introduce the notion of a maximal common Kronecker factor (or quotient) process for two processes. Quasi-disjointness is a notion tied to the homomorphisms from two processes into their maximal common Kronecker factor, and reduces to a previous notion, disjointness, when that factor is trivial. We show that a substantial class of processes, the Weyl processes, are quasi-disjoint from every ergodic process. As a corollary, we show that a Weyl process and an ergodic process are disjoint if and only if they have no nontrivial Kronecker factor in common, or, equivalently, if they form an ergodic product. We give an example which suggests an analogous theory could be constructed in topological dynamics. 0. Introduction. The existence of common eigenvalues other than 1 for ergodic processes implies the existence of nontrivial common factors, and so precludes disjointness. It is known [7] that for certain classes of processes, this is the sole impediment to disjointness, i.e., if the processes have no common eigenvalue other than 1 then the processes are disjoint. We employ a certain mapping from the product of two processes into the Kronecker process which has for its spectrum the eigenvalues common to the two factors to decompose the phase space of the product. The ergodic disintegration of the product measure provides for a measure with certain compatibility properties on almost every one of the pieces. Processes are quasi-disjoint if there is only one such measure on almost all of the pieces. We show that quasi-disjointness is preserved under ergodic group extension and passage to factors. We give two examples, one of which suggests the possibility of an analogous theory in topological dynamics. Presented to the Society, January 23 1970 under the title Quasi-disjointness in ergodic theory, Weyl processes; received by the editors February 26, 1970. AMS 1968 subject classifications. Primary 2670; Secondary 4730, 5482, 6005.