On the classes of Lipschitz and smooth conjugacies of unimodal maps

Waldemar Pa/luba¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 183 (2004) no. 3, pp. 215-227.

Résumé

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 $\alpha >1$.

Détail

DOI : 10.4064/fm183-3-2

Keywords: under mild assumptions lipschitz continuous conjugacy between closures postcritical sets unimodal maps has derivative critical point dense set its preimages restrictive situation infinitely renormalizable maps bounded combinatorial type lipschitz condition automatically implies smoothness conjugacy here critical degree real number alpha