On disjointness of dynamical systems

Abstract

By a dynamical system we mean one of several related objects: measure preserving transformations on probability spaces (processes), self homeomorphisms of compact metric spaces (compact systems), or a combination of these, namely compact systems provided with invariant Borel probability measures. It is the latter, which we call compact processes, which will be of most interest in this paper. In particular, we will study the dynamical properties of the product of two processes with respect to compatible measures – those measures which project to the given measures on the component spaces. This leads to the notion of disjointness of two processes – the only compatible measure is the product measure. As an application we obtain a theorem, a special case of which gives rise to a class of transformations which preserve normal sequences. Finally, we study a topological analog (topological disjointness) and briefly consider the relation between the two notions of disjointness.