Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems

We introduce the multiplicative Ising model and prove basic properties of its ther-modynamic formalism such as existence of pressure and entropies. We generalize to one-dimensional "layer-unique" Gibbs measures for which the same results can be obtained. For more general models associated to a d-dimensional multiplicative invariant potential, we prove a large deviation theorem in the uniqueness regime for averages of multiplicative shifts of general local functions. This thermodynamic formalism is motivated by the statistical properties of multiple ergodic averages.