The spectrum of weakly coupled map lattices

We consider weakly coupled analytic expanding circle maps on the lattice Zd (for d 2 l), with small coupling strength c and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a FrCchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure p* previously obtained by Bricmont-Kupiainen (BKl)) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For d = 1 we also construct Banach spaces of densities with respect to pr on which perturbation theory, applied to the diff…