We study random circle maps that are expanding on the average. Uniform bounds on neither expansion nor distortion are required. We construct a coupling scheme, which leads to exponential convergence of measures (memory loss) and exponential mixing. Leveraging from the structure of the associated correlation estimates, we prove an almost sure invariance principle for vector-valued observables. The motivation for our paper is to explore these methods in a non-uniform random setting. models include [5,6,11,12,16,17,19]. After completing this paper, the authors have also learned of the very recent works [1,9] on the subject, as well as the related [28].The motivation for the paper is twofold. First, we wish to explore the suitability of the coupling method in the above context of non-uniform random maps. Diverting from the papers mentioned, the primary instrument in our analysis is indeed coupling. The coupling method is a soft tool for establishing statistical properties pertaining to the issues of memory loss and correlation decay. In the field of dynamical systems it has been implemented in various works such as [4,7,18,23,25,30] and many others. A transparent introduction to coupling for dynamical systems (in the most elementary setup) can be found in [27]. As to the second motivation, a question that arises naturally is whether other limit laws hold true; we wish to investigate the possibility of proving such laws for the present class of non-uniform random dynamical systems via correlation estimates. It was shown in [8, 20, 22] that a central limit theorem for Sinai billiards follows from correlation bounds involving suitable classes of observables. In [24] a similar approach was taken to prove an almost sure invariance principle (ASIP) for both random and non-random billiard systems. Here we show that, for our system, an ASIP follows from the established correlation estimates with little added work. Yet, the last point is subtle: it depends on the particular form of the correlation estimates, obtained for particular classes of observables. Let us be fully clear that the (averaged) theorems on the Markov chain corresponding to the random maps at issue can certainly be obtained, for example, via spectral methods. Here we present a different approach, which we hope to be of use to other authors beyond the present setup.The paper is organized as follows. In Sec. 2, we introduce the model precisely and record some mathematical preliminaries necessary for understanding the results and the proofs in the rest of the paper. In Sec. 3, we present our main results. In the following Secs. 4-7 we prove these results in the same order as they appear in Sec. 3.
PreliminariesLet S denote the circle obtained by identifying the endpoints of the unit interval [0, 1]. The Lebesgue measure on S 1 is denoted by m.Given α ∈ (0, 1), we denote by C α the set of functions S → R (or S → C) that are Hölder continuous with exponent α. The corresponding Hölder constant is denoted by |f | α . We also introduce the norm