Mixing rates and limit theorems for random intermittent maps

TL;DR

This paper investigates the mixing rates and limit theorems for random intermittent maps, especially Pomeau-Manneville-type, revealing how the fastest mixing component influences the asymptotic behavior of the system.

Contribution

It provides sharp estimates on return times for quenched dynamics and establishes new limit laws, including stable laws and CLT, for random intermittent maps across a broad parameter range.

Findings

Sharp return time estimates for quenched dynamics

Limit laws including CLT and stable laws for $0<\alpha<1$

Existence of infinite invariant measures and correlation asymptotics for $\alpha\geq1$

Abstract

We study random transformations built from 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 around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest $\alpha$) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the \emph{quenched} dynamics. The main applications of this estimate are to \textit{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 detailed in Theorem 3.2. Since our estimates in Theorem 1.1 also hold for $1 \leq \alpha < \infty$ we study a piecewise affine version of our random transformations, prove existence of an infinite ($\sigma-$finite) invariant 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 transformations.