Martingale–coboundary decomposition for families of dynamical systems

Abstract

A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractWe prove statistical limit laws for sequences of Birkhoff sums of the typewhere T n is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T n is replaced by a fixed transformation T , and which is particularly effective in the case when T n varies with n.In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards.As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

“…and linearly interpolate to obtain a process W n ∈ C[0, 1]. The following result is well-known, see for example [19,30,35]:…”

“…and linearly interpolate to obtainx ǫ ∈ C[0, 1]. By [18,Theorem 1,3] (see also [30,Section 6]),x ǫ → w X in C[0, 1] for T nonuniformly expanding of order 2, where X is the solution of the Stratonovich SDE…”

“…In this section, we recall some results of Gordin-type [17] on martingale approximation from [30]. The key advantage of [30] over other martingale approximation methods [17,33,45] is that it gives good control over sums of squares of the approximating martingale, see Proposition 3.5 below.…”

“…Let b = 1/ψ ′ and write z ǫ (n) = ψ(x ǫ (n)),ẑ ǫ (t) = ψ(x ǫ (t)). Definē Proof The assumptions on b ensure that ψ is C 3 uniformly on R. A calculation as in [18,30], using the Taylor expansion of ψ, yields…”

Almost sure invariance principle for random piecewise expanding maps

“…In situations when this method is applicable, it was pointed out in [11] that it gives better error rates in ASIP when compared to those obtained in [17,18]. Finally, we mention the recent important papers by Cuny and Merlevede [4], Korepanov, Kosloff and Melbourne [16], Korepanov [15] as well as Cuny, Dedecker Korepanov and Merlevede [2,3] in which the authors further improved the error rates in ASIP for a wide class of (nonuniformly) hyperbolic deterministic dynamical systems.…”