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

Abstract

For smooth random dynamical systems we consider the quenched linear and higherorder response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gouëzel, Keller, and Liverani [36,32], which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results for random dynamical systems that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasise that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite dimensional multiplicative ergodic theory.

Quenched linear response for smooth expanding on average cocycles

“…For this purpose, we formulate an abstract linear response result for random dynamical systems (see Theorem 11), and afterwards (see Section 4) verify all of its assumptions in the case of parameterized smooth expanding on average cocycles. In sharp contrast with the previously discussed results in [19,23,34], our approach deal with systems exhibiting nonuniform decay of correlations.…”

“…On the other hand, quenched linear response has begun to receive adequate attention only very recently. More precisely, the quenched linear response for (smooth) random dynamical systems was discussed in [34] for expanding dynamics, in [23] for hyperbolic dynamics, and finally in [19] for some classes of partially hyperbolic dynamics.…”

“…The common feature of all those results is that they are restricted to the case when the 'unperturbed' cocycle (T ω,0 ) ω∈Ω exhibits uniform (with respect to ω) decay of correlations (see [34,Remark 4.20], [23, eq. (25)], [19,Definition 3.4] and [19, eq. (QR0)]).…”

“…In particular, our approach requires a detailed analysis of the so-called top Oseledets space of a parameterized transfer operator cocycle (see for example Corollary 35). We stress that unlike [19,23], we do not use the so-called Mather operator, since in our setting it is unclear on which space would this operator act on (due to nonuniform behavior).…”

Statistical stability and linear response for random hyperbolic dynamics

“…A few months after the present paper was made available as a preprint, the Gouëzel-Keller-Liverani theory for cocycles [14] was further generalized in [15], to cover the case of quenched linear, as well as higher-order, response. In particular, [15,Theorem 3.6] generalizes our Theorem 12 to higher-order Taylor expansions, as remarked in [15,Remark 3.8].…”