Linear response for random and sequential intermittent maps

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 independent and identically distributed setting. As an intermediate step, we show existence, uniqueness and statistical stability of the random absolutely continuous invariant probability measure for such nonuniformly 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 Leppänen. 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.