Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails

Abstract

We prove central limit theorems and related results for non-uniformly expanding random dynamical systems and general random Gibbs measures under some integrability conditions and some (relatively weak) mixing related assumptions on the random environment/base map. Some of our conditions also apply to maps which only expand on average, in an appropriate sense. This is the first set of conditions that allows to treat general non-uniformly expanding cases. A crucial step in our proofs can be viewed as effective moderate versions of the random Ruelle-Perron-Frobenius (RPF) theorem for the random transfer operator cocycle corresponding to the random Gibbs measures (namely, effective moderate estimates on the rate of "convergence" to the projection on the top Lyapunov space in the corresponding multiplicative ergodic theorem). As a byproduct of our methods we are also able to prove statistical properties of the corresponding skew products and some estimates on the tails of random mixing times. Other applications like stochastic homogenization and effective linear response formulas will be considered in future projects. Compared with recent progress [40] on the subject, we are able to treat random Gibbs measures corresponding to potentials which do not necessarily have small local oscillation along inverse branches. Moreover, we are able to consider more general classes of random expanding maps with unbounded mixing/covering times and (only) local pairing property. Thus, our results apply to a wide family of C 2 -random expanding maps on compact C 2 manifolds, together with the random geometric potential which corresponds to the unique equivariant family of measures which are absolutely continuous with respect to the volume measure. Some of our results are also new in the case of small local oscillation. Other applications include high dimensional piecewise expanding maps and random subshifts of finite type, which have applications to random Markov interval maps and finite state Markov chains in random dynamical environment.

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

“…Let us also note that, in general, the random variable K(ω) comes from Oseledets theorem and it is not computable. In order to provide explicit conditions for quenched limit 132 Y. Hafouta theorems, in [28] several examples of non-uniformly expanding maps (which are stronger than expansion on average) were given with the property that…”

“…For instance, in [16,[20][21][22] almost sure invariance principle (ASIP, an almost sure approximation by a sum of independent Gaussians) was established for random expanding or hyperbolic maps T ω , in [19,31] Berry-Esseen theorems (optimal rates in the CLT) were obtained for similar classes of maps and in [17,18,23,31] local CLTs were achieved. In addition, in [27] several limit theorems were extended to random non-uniformly hyperbolic or expanding maps. We would also like to refer to [3] for related results concerning mixing rates for random non-uniformly hyperbolic maps and to [32] for related results concerning sequential dynamical systems, where an ASIP was obtained.…”

Quenched limit theorems for expanding on average cocycles

“…Indeed, those are key tools to model many natural phenomena, including the transport in complex environments such as in the ocean or the atmosphere [3]: it is therefore crucial to understand their long term quantitative behavior. Among many remarkable contributions, we particularly emphasize those dealing with the decay of correlations [2,6,7,9,12,15], various (quenched or annealed) limit laws [1,4,16,17,18,19,20,30,31,32,39,43,44,48], as well as recent results devoted to the linear response of random dynamical systems [8,23,47]. For similar results in the closely related context of sequential dynamical systems, we refer to [10,11,33,35,36,37,29,42] and references therein.…”

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding on the average fibers

“…Probably the first papers dealing with quenched limit theorems for random dynamical systems are [36,37], and since then limit theorems for several classes of random dynamical systems were vastly studied. We refer to [3,5,15,16,17,21,22,20,18,28,30,31,51,52] for a partial list of relatively recent results of this kind. We note that in many of the examples these results are obtained for the unique measure µ such that µ ω is absolutely continuous with respect to m. However, some results hold true even for maps T ω : E ω → E σω ⊂ X which are defined on random subsets of X (see [38]), where in this case the most notable choice of µ ω is the, so called, random Gibbs measure (see [28,43]).…”

“…when T ω = T and ϕ(ω, x) = ϕ(x) do not depend on ω, many of the limit theorems follow from spectral properties of the transfer operator L T corresponding to T (namely, the dual of the Koopman operator g → g • T with respect to the underlying invariant measure µ), or from a sufficiently fast convergence of L n T towards a one dimensional projection. While several quenched limit theorems are based on an appropriate random counterpart of such spectral properties (see [15,16,17,21,22,20,18,28,30]), such annealed "spectral" techniques are not fully developed as the quenched ones. A very notable exception is the case of iid maps, discussed in the next section.…”