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We study random transformations built from intermittent maps on the unitinterval that share a common neutral fixed point. We focus mainly on randomselections of Pomeu-Manneville-type maps $T_\alpha$ using the full parameterrange $0< \alpha < \infty$, in general. We derive a number of results around acommon theme that illustrates in detail how the constituent map that is fastestmixing (i.e.\ smallest $\alpha$) combined with details of the randomizingprocess, determines the asymptotic properties of the random transformation. Ourkey result (Theorem 1.1) establishes sharp estimates on the position of returntime intervals for the \emph{quenched} dynamics. The main applications of thisestimate are to \textit{limit laws} (in particular, CLT and stable laws,depending on the parameters chosen in the range $0 < \alpha < 1$) for theassociated skew product; these are detailed in Theorem 3.2. Since our estimatesin Theorem 1.1 also hold for $1 \leq \alpha < \infty$ we study a piecewiseaffine version of our random transformations, prove existence of an infinite($\sigma-$finite) invariant measure and study the corresponding correlationasymptotics. To the best of our knowledge, this latter kind of result iscompletely new in the setting of random transformations.

Mixing rates and limit theorems for random intermittent maps