Fixed points of compositions of earthquakes

19 pages, 1 figure. v2: 21 pages, 3 figures. v2 is a substantial rewrite, with simpler proofs and better explanations, some corrections. v3: further improvements in the expositionLet 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 Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller 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