A Fréchet topology on measured laminations and Earthquakes in the hyperbolic plane

Abstract

We prove that the bijective correspondence between the space of bounded measured laminations $M{L}_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $\lambda \mapsto E^{\lambda }{|}_{S^1}$ is a homeomorphism for the Fréchet topology on $M{L}_b(\mathbb{H})$ and the Teichmüller 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.

31 pages, 7 figures