Decay of geometry for unimodal maps: negative Schwarzian case

Abstract

We show that decay of geometry holds for unimodal maps of the interval which have negative Schwarzian derivative, sufficient finite smoothness, and a nondegenerate critical point. The proof is based on pseudo-analytic extensions of order at least $2$. They allow us to modify Sullivan’s principle that rescaled high iterates of one-dimensional maps tend to analytic limits in such a way that no passage to a limit is actually needed, but the maps are shown to approach the analytic class in a well defined sense. As a technical improvement, this method yields a uniform estimate in the case of renormalizable maps.

Authors

Jacek Graczyk

Laboratoire de Mathématiques d'Orsay
University of Paris XI
91405 Orsay
France

Duncan Sands

Laboratoire de Mathématiques d'Orsay
University of Paris XI
91405 Orsay
France

Grzegorz Świątek

Mathematics Department
Pennylania State University
State College, PA 16802
United States