A Fr\'echet topology on measured laminations and Earthquakes in the hyperbolic plane

Abstract

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichm\"uller space $T(\mathbb{H})$ given by $\lambda\mapsto E^{\lambda}|_{S^1}$ is a homeomorphism for the Fr\'echet topology on $ML_b(\mathbb{H})$ and the Teichm\"uller topology on $T(\mathbb{H})$, where $E^{\lambda}$ is an earthquake with earthquake measure $\lambda$. A corollary is that earthquakes with discrete earthquake measures are dense in $T(\mathbb{H})$. We also establish infinitesimal versions of the above results.