Fixed points of compositions of earthquakes

TL;DR

This paper proves that the composition of two earthquake transformations along filling measured laminations on a surface always has a fixed point in Teichmüller space, linking surface geometry with AdS manifold theory.

Contribution

It establishes the existence of fixed points for compositions of earthquakes along filling laminations, connecting Teichmüller dynamics with AdS geometry.

Findings

Fixed points exist for compositions of earthquakes along filling laminations.

Any two filling measured laminations can be realized as bending laminations of a convex core.

The proof employs geometric estimates from AdS manifold theory.

Abstract

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichm\"uller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichm\"uller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.