Differentiability of the Diffusion Coefficient for a Family of Intermittent Maps

Abstract

It is well known that the Liverani–Saussol–Vaienti map satisfies a central limit theorem for Hölder observables in the parameter regime where the correlations are summable. We show that when C2 observables are considered, the variance of the limiting normal distribution is a C1 function of the parameter. We first show this for the first return map to the base of the second branch by studying the Green-Kubo formula, then conclude the result for the original map using Kac’s lemma and relying on linear response.

A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics

“…This immediately implies the C N−1+η -regularity of the second term of the right-hand side of (45). Moreover, Theorem 3.6 leads to the C N−1+η -regularity of the first term of the right-hand side of (45) via the C N−1+η -regularities of h because (g • h )(ω)(g ,ω ) = h (ω)(g 2 ω ) − h (ω)(g ω ) 2 . This completes the proof.…”

“…In this section, as another application of Theorem 3.6, we show the differentiability of the variances in quenched CLTs for certain class of RDSs (including random Anosov maps and random U(1) extensions of expanding maps). See, e.g., [45] and reference therein for the background of this topic.…”

Linear response for intermittent maps with critical point

“…The remaining integral is exactly the same as (40), and thus to complete the proof of (37) we may repeat the argument for i = 1.…”

“…The map T α,1 is uniformly expanding except at the neutral fixed point at the origin. In this case, robustness of statistical properties have been investigated for both deterministic [6,10,18,27,32,40] and random perturbations [5]. Independently, Baladi and Todd [10] and Korepanov [27] analyzed the regularity properties of the map R ϕ (α) := ´1 0 ϕ dµ α,1 for ϕ ∈ L q with q > 1 sufficiently large, employing different methods.…”