  We study the family of holomorphic maps from the polydisk to the disk which
restrict to the identity on the diagonal. In particular, we analyze the
asymptotics of the orbit of such a map under the conjugation action of a
unipotent subgroup of $\text{PSL}_2(\mathbb{R})$. We discuss an application our
results to the study of the Caratheodory metric on Teichmuller space.
