Linear response for random and sequential intermittent maps

TL;DR

This paper develops a trajectory-wise linear response theory for random and sequential intermittent maps, extending previous averaged results and analyzing the stability and invariant measures of these complex dynamical systems.

Contribution

It introduces a quenched linear response formula for random intermittent maps and investigates the existence, uniqueness, and stability of invariant measures, also highlighting differences in sequential systems.

Findings

Established a quenched linear response formula for random intermittent maps.

Proved existence, uniqueness, and stability of the invariant measure for these systems.

Identified that sequential systems can have infinitely many invariant densities, with only one corresponding to an SRB state.

Abstract

This work establishes a quenched (trajectory-wise) linear response formula for random intermittent dynamical systems, consisting of Liverani-Saussol-Vaienti maps with varying parameters. This result complements recent annealed (averaged) results in the i.i.d setting. As an intermediate step, we show existence, uniqueness and statistical stability of the random absolutely continuous invariant probability measure (a.c.i.m.) for such non-uniformly expanding systems. Furthermore, we investigate sequential intermittent dynamical systems of this type and establish a linear response formula. Our arguments rely on the cone technique introduced by Baladi and Todd and further developed by Lepp{\"a}nen. We also demonstrate that sequential systems exhibit a subtle distinction from both random and autonomous settings: they may possess infinitely many sequential absolutely continuous equivariant densities. However, only one of these corresponds to an SRB state in the sense of Ruelle.