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Transactions of the American Mathematical Society

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Quenched decay of correlations for slowly mixing systems


Authors: Wael Bahsoun, Christopher Bose and Marks Ruziboev
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37A05, 37E05
DOI: https://doi.org/10.1090/tran/7811
Published electronically: May 23, 2019
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Abstract: We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in $ [\alpha _0,\alpha _1]\subset (0,1)$ chosen independently with respect to a distribution $ \nu $ on $ [\alpha _0,\alpha _1]$ and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every $ \delta >0$, for almost every $ \omega \in [\alpha _0,\alpha _1]^\mathbb{Z}$, the upper bound $ n^{1-\frac {1}{\alpha _0}+\delta }$ holds on the rate of decay of correlation for Hölder observables on the fibre over $ \omega $. For three different distributions $ \nu $ on $ [\alpha _0,\alpha _1]$ (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from $ n^{-\frac {1}{\alpha _0}}$ to $ (\log n)^{\frac {1}{\alpha _0}}\cdot n^{-\frac {1}{\alpha _0}}$ to $ (\log n)^{\frac {2}{\alpha _0}}\cdot n^{-\frac {1}{\alpha _0}}$, respectively.


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Additional Information

Wael Bahsoun
Affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom
Email: W.Bahsoun@lboro.ac.uk

Christopher Bose
Affiliation: Department of Mathematics and Statistics, University of Victoria, P.O. BOX 3045 STN CSC, Victoria, British Columbia, V8W 3R4, Canada
Email: cbose@uvic.ca

Marks Ruziboev
Affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom
Email: M.Ruziboev@lboro.ac.uk

DOI: https://doi.org/10.1090/tran/7811
Keywords: Random dynamical systems, slowly mixing systems, quenched decay of correlations.
Received by editor(s): January 29, 2018
Received by editor(s) in revised form: November 19, 2018, and January 17, 2019
Published electronically: May 23, 2019
Additional Notes: The first and third authors would like to thank The Leverhulme Trust for supporting their research through the research grant RPG-2015-346. The second author’s research was supported by a research grant from the National Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2019 American Mathematical Society