Let $\Gamma$ be a finite graph that is symmetric under the interchange of two marked vertices $a$ and $b$. Associated to $\Gamma$ is a rational mapping $R_\Gamma: \mathbb{CP}^2 \dashrightarrow \mathbb{CP}^2$ called the Migdal-Kadanoff Renormalization Mapping. The case that $\Gamma$ is a diamond with $a$ and $b$ being a pair of opposite vertices was studied by Bleher, Lyubich, and the speaker, who showed for this graph that $R_\Gamma$ is partially hyperbolic on an invariant real slice.

We use "twisted versions" of the famous Lee-Yang Theorem from statistical physics and of a theorem by Griffiths and Nishimori on simplicity of Lee-Yang zeros, together with several results from the previous work with Bleher and Lyubich, to prove partial hyperbolicity of $R_\Gamma$ under rather mild assumptions on $\Gamma$ and one technical assumption (which can be easily checked on the computer for any specific graph $\Gamma$). I will also describe how certain special choices of $\Gamma$ lead to mappings with more complicated dynamics. This is work in progress, and we expect our results to lead to a description of the limiting distribution of Lee-Yang zeros for a wide variety of hierarchical lattices.

This is joint work with Olivier Remy.