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

## TL;DR

This paper investigates the asymptotic behavior of holomorphic maps from the polydisk to the disk under group actions, with implications for the Caratheodory metric on Teichmuller space.

## Contribution

It provides new asymptotic analysis of translation flows on holomorphic maps from the poly-plane, linking complex dynamics with Teichmuller theory.

## Key findings

- Characterization of the asymptotic orbit behavior
- Application to the Caratheodory metric on Teichmuller space
- Insights into the dynamics of holomorphic maps under group actions

## 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 $\text{PSL}_2(\mathbb{R})$. We discuss an application our results to the study of the Caratheodory metric on Teichmuller space.