Quenched limit theorems for expanding on average cocycles

Abstract

We prove quenched versions of a central limit theorem, a large deviations principle as well as a local central limit theorem for expanding on average cocycles. This is achieved by building an appropriate modification of the spectral method for nonautonomous dynamics developed by Dragičević et al. (Comm Math Phys 360: 1121-1187, to deal with the case of random dynamics that exhibits nonuniform decay of correlations, which are ubiquitous in the context of the multiplicative ergodic theory. Our results provide an affirmative answer to a question posed by Buzzi (Comm Math Phys 208: 25-54, 1999).

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.…”

“…Quenched statistical stability (i.e. continuity, in a suitable sense, of the map ε → h ω,ε in ε = 0) has been studied for some time: see, e.g., [13] for random subshift of finite type, [9] for smooth expanding maps, [18,23] for Anosov systems. Closer to the focus of the present paper, quenched statistical stability for an expanding on average cocycles of piecewise expanding systems, exhibiting non-uniform decay of correlations (as considered by Buzzi [17]) was established in [25].…”

“…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.…”

“…Our methods rely on the infinite-dimensional version of the multiplicative ergodic theorem (MET) established in [28], coupled with some perturbative estimates on transfer operators generalizing those of [26,27] and close to those in [23]. 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).…”

“…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).…”