  Let $\lambda_-$ and $\lambda_+$ be two bounded measured laminations on the
hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below).
  We consider the left earthquakes along $\lambda_-$ and $\lambda_+$,
considered as maps from the universal Teichm\"uller space $\mathcal T$ to
itself, and we prove that the composition of those left earthquakes has a fixed
point.
  The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism
$u:{\mathbb RP}^1\to {\mathbb RP}^1$, the boundary of the convex hull in
$AdS^3$ of its graph in ${\mathbb RP}^1\times{\mathbb RP}^1\simeq \partial
AdS^3$ is the disjoint union of two embedded copies of the hyperbolic plane,
pleated along measured geodesic laminations. Our main result is that any pair
of bounded measured laminations that "strongly fill" can be obtained in this
manner.
