Linear response for random and sequential intermittent maps

The paper’s Theorem 14 gives a quenched (trajectory-wise) linear response formula for random LSV maps in the regime 0 < α < α < 1/2 with perturbations confined to that compact parameter range, yielding the series with the spatial derivative sitting on Ljσ−jω of XβNβh, with an overall minus sign (their equation (26)) . The proof uses the first-branch transfer operator Nα, the identity that the parameter derivative of the transfer operator produces the distribution (XγNγ(·))′ (no boundary term since the partition is fixed), together with cone estimates that guarantee summability because 1/α − 1 > 1 when α < 1/2 (see (24) and the parameter window (25)) . Lemma 12 provides the needed bounds and zero-average decomposition for (XγNγ(φ))′, matching the integrand structure used in the linear response formula . The candidate solution reproduces the same identity for ∂γLγ and derives the same Neumann-series/resolvent expansion by differentiating the random invariance relation hε(ω)=Lσ−1ω,ε hε(σ−1ω); it then justifies limit exchanges using uniform cone/tower bounds on the compact parameter interval and obtains exactly the series stated in the paper. Minor presentational differences aside (e.g., resolvent-series language vs. the paper’s telescoping-and-integration-in-γ approach), the arguments are essentially the Baladi–Todd cone method adapted to the random setting, and they agree on hypotheses, mechanisms, and the final formula .