Asymptotics of the translation flow on holomorphic maps out of the poly-plane

Abstract

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 PSL 2 ⁡ ( R ) \operatorname {PSL}_2(\mathbb {R}) . We discuss an application of our results to the study of the Carathéodory metric on Teichmüller space.