# The Hilbert metric on Teichm\"uller space and Earthquake

## TL;DR

This paper explores the Hilbert metric on Teichmüller space of punctured surfaces, demonstrating that earthquake rays behave as almost geodesics within this metric framework, based on a specific parametrization.

## Contribution

It introduces the study of the Hilbert metric on Teichmüller space using Hamenstädt's parametrization and proves earthquake rays are almost geodesics in this setting.

## Key findings

- Earthquake rays are almost geodesics under the Hilbert metric.
- The study utilizes Hamenstädt's parametrization of Teichmüller space.
- Provides new insights into the geometric structure of Teichmüller space.

## Abstract

Hamenst\"adt gave a parametrization of the Teichm\"uller space of punctured surfaces such that the image under this parametrization is the interior of a polytope. In this paper, we study the Hilbert metric on the Teichm\"uller space of punctured surfaces based on this parametrization. We prove that every earthquake ray is an almost geodesic under the Hilbert metric.