  In a classical work of the 1950's, Lee and Yang proved that the zeros of the
partition functions of a ferromagnetic Ising models 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 $\RR$ in
two variables (the renormalization transformation). We prove that $\RR$ is
partially hyperbolic on an invariant cylinder $\CC$. The Lee-Yang zeros are
organized 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 critical
temperature, it is $C^\infty$ on a open dense subset, but it vanishes on the
complementary Cantor set of positive measure. This seems to be the first
occasion of a complete rigorous description of the Lee-Yang distributions
beyond 1D models.
