Spatio-temporal large deviations principle for coupled circle maps

We consider the (d +1)-dimensional an dynamical system constituted by weakly coupled expanding circle maps on ℤ^d together with the spatial shifts. This viewpoint allows us to use thermodynamic formalism, and to describe the asymptotic behavior of the system in this setup. We obtain a volume lemma, which describes the exponential behavior of the size under Lebesgue measure of dynamical balls around any orbit, and then a large deviations principle for the empirical measure associated to this dynamical system. The proofs are direct: we do not use the coding constructed by Jiang [Preprint (2002)] for such systems.