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 for such systems.
"Spatio-temporal large deviations principle for coupled circle maps." Ann. Probab. 32 (1B) 692 - 729, January 2004. https://doi.org/10.1214/aop/1079021461