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.