On the classes of Lipschitz and smooth conjugacies of unimodal maps

Under very mild assumptions, any Lipschitz continuous conjugacy between the closures of the postcritical sets of two C-1-unimodal maps has a derivative at the critical point, and also on a dense set of its preimages. In a more restrictive situation of infinitely renormalizable maps of bounded combinatorial type the Lipschitz condition automatically implies the C-1-smoothness of the conjugacy. Here the critical degree can be any real number alpha > 1.