In this paper we construct quasiconformal mappings between Y-pieces so that the corresponding Beltrami coefficient has exponential decay away from the boundary. These maps are used in a companion paper to construct quasiFuchsian groups whose limit sets are non-rectifiable curves of dimension 1.
"Quasiconformal mappings of Y-pieces." Rev. Mat. Iberoamericana 18 (3) 627 - 652, October, 2002.