# 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