Decay of correlations for critically intermittent systems

TL;DR

This paper investigates how randomness influences the statistical properties of intermittent dynamical systems with superattracting fixed points, revealing a phase transition and polynomial decay of correlations.

Contribution

It establishes a phase transition criterion based on randomness parameters and map orders, and shows polynomial decay of correlations even when some deterministic maps have exponential decay.

Findings

A phase transition between finite and infinite invariant measures depending on parameters.

Systems with absolutely continuous invariant measures are mixing.

Correlations decay polynomially despite some maps having exponential decay.

Abstract

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending on the randomness parameters and the orders of the maps at the superattracting fixed point. In case the systems have an absolutely continuous invariant probability measure, we show that the systems are mixing and that the correlations decay polynomially even though some of the deterministic maps present in the system have exponential decay. This contrasts other known results, where random systems adopt the best decay rate of the deterministic maps in the systems.