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.

Improved presentation and results (now only a>1 is needed and consequently $0<{\alpha }_0<1$ in the application for LSV maps)