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K-property for Maharam extensions of non-singular Bernoulli and Markov shifts

Published online by Cambridge University Press:  05 April 2018

ALEXANDRE I. DANILENKO  and
MARIUSZ LEMAŃCZYK
ALEXANDRE I. DANILENKO
Affiliation:
Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Nauky Avenue, Kharkiv, 61103, Ukraine email alexandre.danilenko@gmail.com
MARIUSZ LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland email mlem@mat.umk.pl
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Abstract

It is shown that each conservative non-singular Bernoulli shift is either of type $\mathit{II}_{1}$ or $\mathit{III}_{1}$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Kosloff for equilibrial shifts. Non-equilibrial shifts of type $\mathit{III}_{1}$ are constructed. We further generalize (partly) the main results to non-singular Markov shifts.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3292 - 3321
Copyright
© Cambridge University Press, 2018 

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