Intermittency generated by attracting and weakly repelling fixed points

TL;DR

This paper investigates phase transitions in a class of random dynamical systems with fixed points exhibiting exponential attraction and polynomial repulsion, extending previous results to a new setting.

Contribution

It introduces a new family of random systems with different fixed point dynamics and proves the existence of a phase transition using a novel method involving invariant sets.

Findings

Phase transition persists under different fixed point dynamics.

Construction of invariant sets is key to the proof.

Results extend understanding of intermittency in random systems.

Abstract

Recently for a class of critically intermittent random systems a phase transition was found for the finiteness of the absolutely continuous invariant measure. The systems for which this result holds are characterized by the interplay between a superexponentially attracting fixed point and an exponentially repelling fixed point. In this article we consider a closely related family of random systems with instead exponentially fast attraction to and polynomially fast repulsion from two fixed points, and show that such a phase transition still exists. The method of the proof however is different and relies on the construction of a suitable invariant set for the transfer operator.