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Ergodic cocycles of IDPFT systems and non-singular Gaussian actions

Part of: Ergodic theory

Published online by Cambridge University Press:  18 February 2021

ALEXANDRE I. DANILENKO Open the ORCID record for ALEXANDRE I. DANILENKO [Opens in a new window]  and
MARIUSZ LEMAŃCZYK
ALEXANDRE I. DANILENKO*
Affiliation:
B. I. Verkin Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv61103, Ukraine
MARIUSZ LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100Toruń, Poland (e-mail: mlem@mat.umk.pl)
*
e-mail: alexandre.danilenko@gmail.com
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Abstract

It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of non-singular infinite direct products T of transformations $T_n$ , $n\in \mathbb N$ , of finite type is studied. It is shown that if $T_n$ is mildly mixing, $n\in \mathbb N$ , the sequence of Radon–Nikodym derivatives of $T_n$ is asymptotically translation quasi-invariant and T is conservative then the Maharam extension of T is sharply weak mixing. This technique provides a new approach to the non-singular Gaussian transformations studied recently by Arano, Isono and Marrakchi.

Keywords

ergodic transformationGaussian cocycleMaharam extension

MSC classification

Primary: 37A40: Nonsingular (and infinite-measure preserving) transformations
Secondary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 37A50: Relations with probability theory and stochastic processes
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022 , pp. 1624 - 1654
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

To the memory of Sergiy Sinel’shchikov, our colleague and friend

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