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

Tim Austin¹

¹ Department of Mathematics University of California Los Angeles, CA 90095-1555, U.S.A.

Fundamenta Mathematicae, Tome 210 (2010) no. 2, pp. 133-206.

Résumé

We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to 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, of the Furstenberg–Zimmer structure theory and of results of Mentzen describing the structure of automorphisms of such extensions when they are relatively ergodic. 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 non-conventional ergodic averages. Although both applications are modest in themselves, we hope that they point towards a broader usefulness of this formalism in ergodic theory.

Détail

DOI : 10.4064/fm210-2-3

Keywords: study generalized notion homogeneous skew product extension probability preserving system which homogeneous space fibres allowed vary ergodic decomposition base construction extensions rests simple notion direct integral measurable family homogeneous spaces which has number precedents older literature main contribution present paper systematic development formalism handling extensions including non ergodic versions results mackey describing ergodic components extensions furstenberg zimmer structure theory results mentzen describing structure automorphisms extensions relatively ergodic offer applications structural results actions several commuting transformations firstly describing possible joint distributions three isotropy factors corresponding three commuting transformations secondly describing characteristic factors system double non conventional ergodic averages although applications modest themselves hope point towards broader usefulness formalism ergodic theory