Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications

We study a generalized notion of a homogeneous skew-product extension of a probability-preserving base system in which the homogeneous space fibres can vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of ‘direct integral’ for a ‘measurable family’ of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions, including non-ergodic versions of the results of Mackey describing ergodic components of such extensions [29], of the Furstenberg-Zimmer Structure Theory [45, 44, 18] and of results of Mentzen [32] describing the structure of automorphisms of relatively ergodic such extensions. We then offer applications to two structural results for actions of several commuting transformations: firstly to describing the possible joint distributions of three isotropy factors corresponding to three commuting transformations; and secondly to describing the characteristic factors for a system of double nonconventional ergodic averages (see [3] and the references listed there). Although both applications are modest in themselves, we hope that they point towards a broader usefulness of this formalism in ergodic theory.