Almost sure rates of mixing for random intermittent maps

TL;DR

This paper studies the statistical properties of a family of intermittent maps with neutral fixed points, showing that random compositions exhibit specific decay rates of correlations using a generalized tower construction.

Contribution

It extends the random tower construction to a family of maps with variable tangency orders, enabling analysis of decay of correlations in this setting.

Findings

Random compositions have decay of correlations of order n^{1-1/α_0+δ}.

The generalized tower construction applies to maps without a common Markov partition.

Upper bounds for quenched decay of correlations are established.

Abstract

We consider a family $\mathcal F$ of maps with two branches and a common neutral fixed point $0$ such that the order of tangency at $0$ belongs to some interval $[\alpha_0, \alpha_1]\subset (0, 1)$. Maps in $\mathcal F$ do not necessarily share a common Markov partition. At each step a member of $\mathcal F$ is chosen independently with respect to the uniform distribution on $[\alpha_0, \alpha_1]$. We show that the construction of the random tower in Bahsoun-Bose-Ruziboev \cite{BBR} with \emph{general return time} can be carried out for random compositions of such maps. Thus their general results are applicable and gives upper bounds for the quenched decay of correlations of form $n^{1-1/\alpha_0+\delta}$ for any $\delta>0$.