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

Abstract

We extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving [9] to quenched random piecewise hyperbolic dynamics including some classes of billiards. For general ergodic sequences of maps in a neighbourhood of a hyperbolic map we prove a quenched large deviations principle (LDP), central limit theorem (CLT), and local central limit theorem (LCLT).

Almost sure invariance principle for random piecewise expanding maps

“…13.] (which are precisely the same as in [8,Proposition 4.4.]). However, we stress that the corresponding almost sure invariance principle result (when restricted to scalar-valued observables) is weaker than the one established in [6, Theorem 1.…”

“…Remark 1. It was pointed out in [8,Section 3] that under our assumptions, it is possible to apply the multiplicative ergodic theorem to the cocycle (L ω ) ω∈Ω . Let Y 1 (ω) denotes the Oseledets subspace corresponding that corresponds to its largest Lyapunov exponent (which is 0).…”

“…It turns out that Y 1 (ω) is spanned by h 0 ω (see [8, Proposition 3.6.]). We refer to [8] for more details.…”

“…ω ∈ Ω, n ∈ N and t ∈ R d such that |t| ≤ ρ. Indeed, (23) was proved in [8] in the case when d = 1, i.e. when g is a real-valued observable.…”

Quadratic response of random and deterministic dynamical systems

“…In this approach, one studies the response of an appropriate equivariant family of measure to the perturbations. The interest of this approach was highlighted in the climate literature (notably [11]), but so far the mathematical results in this direction are very sparse: see [32] where the problem of quenched response in the context of random products of uniformly expanding maps is studied, or [12] where the same problem is studied for random product of (close-by) Anosov diffeomorphisms. Linear request, optimal response, numerical methods.…”