Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder

In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal–Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C. The Lee–Yang zeros are organized in a transverse measure for the central-stable foliation of R|C. Their distribution is absolutely continuous. Its density is C^∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C^∞ on a open dense subset, but it vanishes on the complementary set of positive measure.