Quenched Linear Response for Smooth Expanding on Average Cocycles

Almost sure invariance principle for random dynamical systems via Gouëzel's approach

“…Our results have two major advantages when compared to [17]. Firstly, we show that our class of examples exhibits both quenched and annealed linear response.…”

“…We emphasize that all three papers deal with cases of random dynamics which exhibit uniform decay of correlations (with respect to the random parameter ω ∈ Ω). More recently, Dragičević, Giulietti and Sedro [17] established the quenched linear response for a class of random dynamics which exhibits nonuniform decay of correlations. More precisely, they considered the case of cocycles which are expanding on average.…”

“…More precisely, they considered the case of cocycles which are expanding on average. Namely, in [17] it is assumed that there exists a log-integrable random variable γ : Ω → (0, ∞) such that γ ω,ε ≥ γ and Ω log γ(ω) dP(ω) > 0, (1) where γ ω,ε denotes the minimal expansion of T ω,ε . Note that (1) allows for γ ω,ε < 1 on a set of positive measure.…”

“…Thus, in sharp contrast to [34], it is not required that all maps T ω,ε are expanding or that there exists a uniform (in ω) lower bound for the minimal expansion. The main result of [17] yields that for each ε, there is a measurable family (h ω,ε ) ω∈Ω lying in the Sobolev space W 3,1 such that:…”

Linear response for intermittent maps with critical point

“…There has been deep investigation about response in the setting of unimodal maps [2,7,9,37,39], providing examples of systems where the SRB measure does not vary smoothly under perturbations. Recently, linear response has also been actively studied in the context of random and extended dynamical systems; see, for instance [13,17,19,25,41]. For extensive background information on linear response, we recommend the survey [8].…”