A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems

Abstract

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form T σ n−1 ω • · · · • T σω • T ω . An important aspect of our results is that we only assume ergodicity and invertibility of the random driving σ : Ω → Ω; in particular no expansivity or mixing properties are required.

A spectral approach for quenched limit theorems for random hyperbolic dynamical systems

“…In our previous paper [9] we extended the Nagaev-Guivarc'h spectral method to obtain limit theorems, such as the Central Limit Theorem (CLT), the Large Deviation Principle (LDP) and the Local Central Limit Theorem (LCLT), for random dynamical systems governed by a cocycle of maps T (n) ω := T σ n−1 ω • · · · • T σω • T ω , assuming uniform-in-ω eventual expansivity conditions on the maps T ω . The random driving was a general ergodic, invertible transformation σ : Ω on a probability space (Ω, P), and the real observable g was defined on the product space Ω × X → R.…”

“…9 Proof. One can follow the proof of Lemma 3.3 (part 2) [9], using the definition of the untwisted transfer operator (7) and Lemma 2.2.…”

“…In the quenched random setting we must replace the n-th power of the twisted operator with the twisted transfer operator cocycle L θ,(n) ω := L θ σ n−1 ω • · · · • L θ σω • L θ ω . By using the multiplicative ergodic theorem adapted to the study of such cocycles and generalizing a theorem of Hennion and Hérve [18] to the random setting, we were able in our previous paper [9] to show that the cocycle L θ,(n) ω is quasi-compact for θ near to 0. We therefore thus obtained that for such values of θ and for P-a.e.…”

“…Specifically, in the smooth hyperbolic setting and in any dimension, we use the functional analytic setup of Gouëzel and Liverani [15], and in the piecewise hyperbolic case in dimension two we use the spaces from Demers and Liverani [5] (as well as Demers and Zhang [6,7]). This increased technicality in the underlying spaces necessitates a certain amount of checking of relevant conditions, however, we wish to highlight the fact that a wholesale change of the theory of [9] is not required, which demonstrates the power and flexibility of our approach. The use of transfer operators in the study of statistical properties and limit theorems for hyperbolic dynamical systems has flourished in the last years, and [2] presents a thorough discussion of the various spaces that have been used in the literature.…”

Almost sure invariance principle for random piecewise expanding maps

“…• maps T ω , ω ∈ Ω are Anosov diffeomorphisms on a compact Riemannian manifold M that belong to a sufficiently small neighborhood of a fixed Anosov diffeomorphism T on M; • maps T ω , ω ∈ Ω are suitable perturbations of a billiard map associated to the periodic Lorentz gas studied by Demers and Zhang [5]; • (T ω ) ω∈Ω is a family of piecewise expanding maps (either on the unit interval or in higher dimension) satisfying appropriate conditions as in [6,7].…”