Linear response for random and sequential intermittent maps

TL;DR

The paper develops trajectory-wise (quenched) linear response theory for random and sequential intermittent maps in the LSV family, establishing both statistical stability of random acims and a detailed linear response formula under small perturbations. The approach hinges on cone techniques (Baladi–Todd, Leppänen) to obtain uniform regularity and distortion control, enabling a differentiable dependence of random and sequential invariant densities on perturbations. It shows that random intermittent systems admit a unique random acim up to measurability and ergodicity, while sequential dynamics may exhibit non-uniqueness of sequential acims, with only the SRB-type state in the cone capturing the physical limit from the infinite past. The results extend to $L^q$ observables with appropriate weighting, broadening applicability to more general measurable outputs and connecting quenched responses to annealed findings in the i.i.d. setting.

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{ä}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.

Theorems & Definitions (42)

• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 4
• Proposition 5
• Remark 6
• proof : Proof of Proposition \ref{['prop:conecontraction']}
• Theorem 7