### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 133-206 |

Number of pages | 74 |

Journal | Fundamenta Mathematicae |

Volume | 210 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Ergodic decomposition
- Furstenberg-Zimmer theory
- Isometric extension
- Mackey theory
- Probability-preserving system
- Skew product

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*210*(2), 133-206. https://doi.org/10.4064/fm210-2-3

**Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications.** / Austin, Tim.

Research output: Contribution to journal › Article

*Fundamenta Mathematicae*, vol. 210, no. 2, pp. 133-206. https://doi.org/10.4064/fm210-2-3

}

TY - JOUR

T1 - Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications

AU - Austin, Tim

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Ergodic decomposition

KW - Furstenberg-Zimmer theory

KW - Isometric extension

KW - Mackey theory

KW - Probability-preserving system

KW - Skew product

UR - http://www.scopus.com/inward/record.url?scp=84855892284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855892284&partnerID=8YFLogxK

U2 - 10.4064/fm210-2-3

DO - 10.4064/fm210-2-3

M3 - Article

VL - 210

SP - 133

EP - 206

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 2

ER -