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In a classical work of the 1950's, Lee and Yang proved that the zeros of thepartition functions of a ferromagnetic Ising models always lie on the unitcircle. Distribution of these zeros is physically important as it controlsphase transitions in the model. We study this distribution for theMigdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can bedescribed in terms of the dynamics of an explicit rational function $\RR$ intwo variables (the renormalization transformation). We prove that $\RR$ ispartially hyperbolic on an invariant cylinder $\CC$. The Lee-Yang zeros areorganized in a transverse measure for the central-stable foliation of $\RR|\,\CC$. Their distribution is absolutely continuous. Its density is $C^\infty$(and non-vanishing) below the critical temperature. Above the criticaltemperature, it is $C^\infty$ on a open dense subset, but it vanishes on thecomplementary Cantor set of positive measure. This seems to be the firstoccasion of a complete rigorous description of the Lee-Yang distributionsbeyond 1D models.

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