Associated to any finite simple graph \(\Gamma = (V, E)\) is the chromatic polynomial \(P_\Gamma(q)\), which has the property that for any integer \(k \geq 0\), \(P_\Gamma (k)\) is the number of proper coloring of \(\Gamma\). A hierarchical lattice is a sequence of graphs \(\{\Gamma_n\}_{n=0}^\infty\) built recursively under a generating graph. For each \(n \geq 0\), let \(\mu_n\) be the probability measure

\[\begin{equation*} \mu_n:=\frac{1}{|V_n|} \sum_{\substack{q \in \mathbb{C} \\ P_{\Gamma_n}(q)=0}} \delta_q. \end{equation*}\]

We prove that if the generating graph is 2-connected, then the sequence of measures \(\mu_n\) converges to some measure \(\mu\), called the limiting measure of chromatic zeros for \(\{\Gamma_n\}_{n=0}^\infty\). For the Diamond Hierarchical Lattice (DHL), we show that its limiting measure has Hausdorff dimension 2.

The main techniques come from holomorphic dynamics and arithmetic dynamics. In particular we prove a new equidistribution result that relates the chromatic zeros of a hierarchical lattice to the bifurcation/activity current associated to a particular marked point. This is joint work with Roland Roeder.