Under very mild assumptions, any Lipschitz continuous conjugacy between the closures of the postcritical sets of two C¹-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 C¹-smoothness of the conjugacy. Here the critical degree can be any real number α > 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-3-2,
author = {Waldemar Pa\l uba},
title = {On the classes of Lipschitz and smooth conjugacies of unimodal maps},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {215-227},
zbl = {1067.37050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-3-2}
}
Waldemar Pałuba. On the classes of Lipschitz and smooth conjugacies of unimodal maps. Fundamenta Mathematicae, Tome 184 (2004) pp. 215-227. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-3-2/