  We investigate the location of zeros for the partition function of the
anti-ferromagnetic Ising Model, focusing on the zeros lying on the unit circle.
We give a precise characterization for the class of rooted Cayley trees,
showing that the zeros are nowhere dense on the most interesting circular arcs.
In contrast, we prove that when considering all graphs with a given degree
bound, the zeros are dense in a circular sub-arc, implying that Cayley trees
are in this sense not extremal. The proofs rely on describing the rational
dynamical systems arising when considering ratios of partition functions on
recursively defined trees.
