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Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles


Authors: Jean-Pierre Conze, Tomasz Downarowicz and Jacek Serafin
Journal: Trans. Amer. Math. Soc. 369 (2017), 3421-3441
MSC (2010): Primary 37A05, 37A20; Secondary 37A45
DOI: https://doi.org/10.1090/tran/6788
Published electronically: September 13, 2016
MathSciNet review: 3605976
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Abstract: The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shift-invariant measures supported by the symbolic space with complex symbols. We also prove a ``lifting lemma'' for generic points: given a joining $ \xi $ of two shift-invariant measures $ \mu $ and $ \nu $, every point $ x$ generic for $ \mu $ lifts to a pair $ (x,y)$ generic for $ \xi $ (such $ y$ exists in the full symbolic space). This lemma allows us to translate correlation between bounded sequences to the language of correlation of measures. Finally, to establish that the property of an invariant measure being uncorrelated to any ergodic measure is essentially weaker than the property of being disjoint from any ergodic measure, we develop and apply criteria for ergodicity of four-jump cocycles over irrational rotations. We believe that apart from the applications to studying the notion of correlation, the two developed tools, the lifting lemma and the criteria for ergodicity of four-jump cocycles, are of independent interest. This is why we announce them also in the title. In the Appendix we also introduce the notion of conditional disjointness.


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Additional Information

Jean-Pierre Conze
Affiliation: IRMAR, CNRS UMR 6625, University of Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France

Tomasz Downarowicz
Affiliation: Institute of Mathematics, Polish Academy of Science, Śniadeckich 8, 00-656 Warsaw, Poland

Jacek Serafin
Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

DOI: https://doi.org/10.1090/tran/6788
Received by editor(s): February 9, 2015
Received by editor(s) in revised form: May 8, 2015
Published electronically: September 13, 2016
Additional Notes: The research of the second and third authors was supported by the NCN (National Science Center, Poland) grant 2013/08/A/ST1/00275.
Article copyright: © Copyright 2016 American Mathematical Society