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.