Quenched decay of correlations for slowly mixing systems

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.