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

100%
Austin T.
Fundamenta Mathematicae
|
2010
|
tom 210
|
nr 2
133-206
EN
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.
2
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The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

100%
Austin T.
Analysis and Geometry in Metric Spaces
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2016
|
tom 4
|
nr 1
EN
Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.
3
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Entropy of probability kernels from the backward tail boundary

100%
Austin T.
Studia Mathematica
|
2015
|
tom 227
|
nr 3
249-257
EN
A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space (X,μ). These works have culminated in a proof by Downarowicz and Frej that various competing definitions all coincide, and that the resulting quantity is uniquely characterized by certain abstract properties. On the other hand, Makarov has shown that this 'operator entropy' is always dominated by the Kolmogorov-Sinai entropy of a certain classical system that may be constructed from a Markov operator, and that these numbers coincide under certain extra assumptions. This note proves that equality in all cases.
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