# Geodesic length spectrum of hyperelliptic connected components

## TL;DR

This paper introduces a new framework for analyzing pseudo-Anosov homeomorphisms on translation surfaces, enabling the computation of systoles in hyperelliptic components and describing their length spectrum without computational aid.

## Contribution

It provides a novel, computer-free method to compute systoles and analyze the length spectrum of hyperelliptic components in Teichmüller dynamics.

## Key findings

- Computed systoles for hyperelliptic components
- Described the bottom of the length spectrum
- Established a general framework for pseudo-Anosov analysis

## Abstract

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmueller geodesic flow restricted to the hyperelliptic connected components, settling a question of Farb. We stress that all proofs and computations are performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components.