  Associated to any finite simple graph $\Gamma$ is the chromatic polynomial
$P_\Gamma(q)$ whose complex zeroes are called the chromatic zeros of $\Gamma$.
A hierarchical lattice is a sequence of finite simple graphs
$\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule
expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the
probability measure that assigns a Dirac measure to each chromatic zero of
$\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the
sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We
call $\mu$ the limiting measure of chromatic zeros associated to
$\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we
prove that the support of $\mu$ has Hausdorff dimension two.
  The main techniques used come from holomorphic dynamics and more specifically
the theories of activity/bifurcation currents and arithmetic dynamics. We prove
a new equidistribution theorem that can be used to relate the chromatic zeros
of a hierarchical lattice to the activity current of a particular marked point.
We expect that this equidistribution theorem will have several other
applications.
