Mixing rates and limit theorems for random intermittent maps

Wael Bahsoun, Christopher Bose

Published 2015-03-20Version 1

We study random intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_\alpha$ using the full parameter range $0< \alpha < \infty$, in general. We derive a number of results under a common theme illustrating how the constituent map that is fastest mixing (i.e. smallest $\alpha$) dominates the asymptotic properties of the random map. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the quenched dynamics. The main applications of our estimate are to limit laws (in particular, CLT and stable laws, depending on the parameters chosen in the range $0 < \alpha < 1$) for the associated skew product; these are treated in Theorem 3.2. Since our estimates also hold for $1 \leq \alpha < \infty$ we also study a second class of random transformations derived from piecewise affine Gaspard-Wang maps, prove existence of an infinite ($\sigma-$finite) measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random maps.