On the homeomorphisms of the space of geodesic laminations on a   hyperbolic surface
  We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface. 
Charalampos Charitos, Ioannis Papadoperakis, Athanase Papadopoulos   (IRMA)
math.GT