Se p 20 06 LIMIT THEOREMS FOR COUPLED INTERVAL MAPS

We prove a local limit theorem for Lipschitz continuous observables on a weakly coupled lattice of piecewise expanding interval maps. The core of the paper is a proof that the spectral radii of the Fourier-transfer operators for such a system are strictly less than 1. This extends the approach of [KL06] where the ordinary transfer operator was studied. 1. Results This paper deals with the issue of probabilistic limit theorems in dynamical systems, i.e., limit theorems for the Birkhoff sums Snf = ∑n−1 k=0 f ◦ T , where T is a probability preserving transformation of a space X and f : X → R is an appropriate measurable function. There are currently many techniques available to prove the central limit theorem Snf/ √ n → N (0, σ), let us mention for example elementary techniques, martingales, spectral arguments. On the other hand, if one is interested in the local limit theorem μ{Snf ∈ [a, b]} ∼ |b−a| σ2πn , the scope of possible techniques is much more narrow: all known proofs rely on spectral analysis of transfer operators. Therefore, the class of systems for which a local limit theorem is proved is much smaller. We are interested in limit theorems for coupled map lattices. The only previous result in this context is [Bar02], where central limit theorem, moderate deviations principle and a partial large deviations principle were established under strong analyticity assumptions on the local map and the coupling. In this paper, we establish central and local limit theorems for coupled interval maps under much weaker assumptions. More precisely, we study the same class of systems as in [KL06]. We emphasize on local limit theorem, since it is the most demanding result. But our method, relying on spectral analysis of transfer operators, gives other limit theorems, see Remark 1.4 below. Let us recall the setup from [KL06]. Given a compact interval I ⊂ R we will consider the phase space Ω := I d . In the following we always assume without loss of generality that I = [0, 1]. The single site dynamics is given by a map τ : I → I. We assume τ to be a continuous, piecewise C2 map from I to I with singularities at ζ1, . . . , ζN−1 ∈ (0, 1) in the sense that τ is monotone and C2 on each component of I \ {ζ0 = Date: February 2, 2008. 2000 Mathematics Subject Classification. 37L60,60F05.