Quenched decay of correlations for slowly mixing systems (1706.04158v3)

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 H\"older 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.