A cyclic extension of the earthquake flow

TL;DR

This paper introduces a new circle action on Teichmüller space that extends the earthquake flow, providing a broader framework with properties analogous to classical earthquake theory and connections to complex earthquakes.

Contribution

It constructs a cyclic extension of the earthquake flow on Teichmüller space, generalizing its properties and extending to universal Teichmüller space.

Findings

Defines a circle action on Teichmüller space that limits to earthquake flow.

Extends Thurston's Earthquake Theorem to this new action.

Connects to complex earthquakes and universal Teichmüller space.

Abstract

Let $\cT$ be Teichm\"uller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the main properties of the earthquake flow, for instance it satisfies an extension of Thurston's Earthquake Theorem and it has a complex extension which is analogous and limits to complex earthquakes. Moreover, a related circle action on $\cT\times \cT$ extends to the product of two copies of the universal Teichm\"uller space.