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 <!-- MATH $[\alpha _0,\alpha _1]\subset (0,1)$ --> chosen independently with respect to a distribution on <!-- MATH $[\alpha _0,\alpha _1]$ --> 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 0$">, for almost every <!-- MATH $\omega \in [\alpha _0,\alpha _1]^\mathbb{Z}$ --> , the upper bound <!-- MATH $n^{1-\frac {1}{\alpha _0}+\delta }$ --> holds on the rate of decay of correlation for Hölder observables on the fibre over . For three different distributions on <!-- MATH $[\alpha _0,\alpha _1]$ --> (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from <!-- MATH $n^{-\frac {1}{\alpha _0}}$ --> to <!-- MATH $(\log n)^{\frac {1}{\alpha _0}}\cdot n^{-\frac {1}{\alpha _0}}$ --> to <!-- MATH $(\log n)^{\frac {2}{\alpha _0}}\cdot n^{-\frac {1}{\alpha _0}}$ --> , respectively.

We prove statistical stability for a family of Lorenz attractors with a stable foliation.

Critical intermittency stands for a type of intermittent dynamics in iterated function systems, caused by an interplay of a superstable fixed point and a repelling fixed point. We consider critical intermittency for iterated function systems of interval maps and demonstrate the existence of a phase transition when varying probabilities, where the absolutely continuous stationary measure changes between finite and infinite. We discuss further properties of this stationary measure and show that its density is not in $$L^q$$L q for any $$q>1$$q > 1 . This provides a theory of critical intermittency alongside the theory for the well studied Manneville–Pomeau maps, where the intermittency is caused by a neutral fixed point.

The classical Lorenz flow, and any flow which is close to it in the $$C^2$$C2 -topology, satisfies a Central Limit Theorem (CLT). We prove that the variance in the CLT varies continuously.