Best Lipschitz maps and Earthquakes

TL;DR

This paper explores the relationship between best Lipschitz maps, transverse measures, and earthquakes on hyperbolic surfaces, establishing a duality and detailed correspondence among these concepts.

Contribution

It introduces a Lie algebra valued measure framework and demonstrates that these measures are infinitesimal earthquakes, deepening the understanding of Thurston's duality conjecture.

Findings

Lie algebra valued measures correspond to transverse measures

These measures are shown to be infinitesimal earthquakes

Establishes a natural correspondence between best Lipschitz maps and earthquakes

Abstract

This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces (infinity harmonic maps), which induce dual Lie algebra valued transverse measures with support on Thurston's canonical lamination. The present paper examines these Lie algebra valued measures in greater detail. For any measured lamination we are led to define a Lie algebra valued measure and conversely every Lie algebra valued transverse measure arrises from this process. Furthermore, we show that such measures are infinitesimal earthquakes. This construction provides a natural correspondence between best Lipschitz maps and earthquakes.