In a classical work of the 1950's, Lee and Yang proved that for fixed
nonnegative temperature, the zeros of the partition functions of a
ferromagnetic Ising model always lie on the unit circle in the complex magnetic
field. Zeros of the partition function in the complex temperature were then
considered by Fisher, when the magnetic field is set to zero. Limiting
distributions of Lee-Yang and of Fisher zeros are physically important as they
control phase transitions in the model. One can also consider the zeros of the
partition function simultaneously in both complex magnetic field and complex
temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF)
zeros. In this paper we continue studying their limiting distribution for the
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
Migdal-Kadanoff renormalization transformation). We study properties of the
Fatou and Julia sets of this transformation and then we prove that the
Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical
(1,1)-current in the projective space. The free energy of the lattice gets
interpreted as the pluripotential of this current. We also prove a more general
equidistribution theorem which applies to rational mappings having
indeterminate points, including the Migdal-Kadanoff renormalization
transformation of various other hierarchical lattices.Comment: Continues arXiv:1009.4691. Final version. To appear in The Journal of
Geometric Analysis. (This is a tiny update to correct an error in the Latex
file from previous version.
