  We introduce the multiplicative Ising model and prove basic properties of its
thermodynamic 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.
