# Algebra/Number Theory Seminar

## EQUIDISTRIBUTION OF CHROMATIC ZEROS AND ARITHMETIC DYNAMICS

Ivan Chio, UR

Thursday, September 10th, 2020
2:00 PM - 3:00 PM
https://rochester.zoom.us/j/92510657759

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

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.

Event contact: c dot d dot haessig at rochester dot edu