Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

Abstract

We consider the problem of stability and approximability of Oseledets splittings and Lyapunov exponents for Perron-Frobenius operator cocycles associated to random dynamical systems. By developing a random version of the perturbation theory of Gouëzel, Keller, and Liverani, we obtain a general framework for solving such stability problems, which is particularly well adapted to applications to random dynamical systems. We apply our theory to random dynamical systems consisting of C k expanding maps on S 1 (k ≥ 2) and provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the associated Perron-Frobenius operator cocycle to (i) uniformly small fiberwise C k−1 -perturbations to the random dynamics, and (ii) numerical approximation via a Fejér kernel method. A notable addition to our approach is the use of Saks spaces, which provide a unifying framework for many key concepts in the so-called 'functional analytic' approach to studying dynamical systems, such as Lasota-Yorke inequalities and Gouëzel-Keller-Liverani perturbation theory.

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

“…Remark 3.9. The claim that 'there exists ǫ 0 ∈ (0, 1] such that (A ǫ , σ) is ξ-mixing whenever |ǫ| < ǫ 0 ' is exactly the content of [20, Proposition 1] (as well as being an easy corollary of [16,Proposition 3.11]). Upon examining these proofs it is clear that something slightly stronger is true: in the setting of Theorem 3.6, for every κ ∈ (ρ, 1) there exists ǫ κ > 0 such that for all ǫ ∈ (−ǫ κ , ǫ κ ) one has sup…”

“…On the other hand, the literature on quenched response theory for random dynamical systems is relatively small, and has only recently become an active research topic. With a few notable exceptions, most results for random systems have focussed on the continuity of the equivariant random measure [8,3,24,28,40], although some more generally apply to the continuity of the Oseledets splitting and Lyapunov exponents associated to the RDS's Perron-Frobenius operator cocycle [11,16]. Quenched linear and higher-order response results are, to the best of our knowledge, limited to [47], where quenched linear and higher-order response is proven for general RDSs of C k uniformly expanding maps, and to [20], wherein quenched linear response is proven for RDSs of Anosov maps nearby a fixed Anosov map.…”

Quenched linear response for smooth expanding on average cocycles

“…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].…”

Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach

“…In [11], the authors investigate connections of this phenomenon with escape rates from random sets. Very recently, there has been some further progress: In [16] and [17], Horan provides bounds on the second Lyapunov exponent for a class of random interval maps using cone techniques, and in [4], Crimmins shows a result on stability for hyperbolic Oseledets splittings for quasi-compact cocycles, modelled on the famous stability result of Keller and Liverani [20].…”