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.
"Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems." Electron. J. Probab. 19 1 - 19, 2014. https://doi.org/10.1214/EJP.v19-3189