Cusp excursions for the earthquake flow on the once-punctured torus

TL;DR

This paper investigates the behavior of earthquake trajectories on the once-punctured torus, revealing differences from horocyclic flow during cusp excursions and establishing a link between systole and continued fractions.

Contribution

It demonstrates that earthquake flow exhibits distinct cusp excursion behavior compared to horocyclic flow and connects systole functions with continued fractions.

Findings

Earthquake flow and horocyclic flow behave differently in cusp excursions.

A relation between systole function and continued fractions is established.

The study provides insights into the dynamics of earthquake trajectories on moduli space.

Abstract

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow and the horocyclic flow behave very differently in cusp excursions. In particular, we prove a relation between the systole function and continued fractions and discuss the cusp excursions of earthquake trajectories.