Quenched decay of correlations for slowly mixing systems
journal contributionposted on 15.11.2019 by Wael Bahsoun, Christopher Bose, Marks Ruziboev
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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 LiveraniSaussol-Vaienti maps with parameters in [α0, α1] ⊂ (0, 1) chosen independently with respect to a distribution ν on [α0, α1] 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 δ > 0, for almost every ω ∈ [α0, α1] Z, the upper bound n 1− 1 α0 +δ holds on the rate of decay of correlation for Holder observables on the fibre over ¨ ω. For three different distributions ν on [α0, α1] (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from n − 1 α0 to (log n) 1 α0 · n − 1 α0 to (log n) 2 α0 · n − 1 α0 respectively.
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WB and MR would like to thank The Leverhulme Trust for supporting their research through the research grant RPG-2015-346. CB’s research is supported by a research grant from the National Sciences and Engineering Research Council of Canada.
- Mathematical Sciences