Planar embeddings of inverse limit spaces of unimodal maps

For an arbitrary Cr unimodal map f, its inverse limit space X is embedded in a planar region W in such a way that the induced homeomorphism f̂:X→X conjugates to a Lipschitz map g:A⊂W→A. Furthermore g can be extended, in a Cr manner, to the rest of W. In the course of the proof a characterization of the endpoints of X in terms of symbolic dynamics is given.