Distortion bounds for $C^{2+\eta}$ unimodal maps

Mike Todd¹

¹ Department of Mathematics University of Surrey Guildford, Surrey, GU2 7XH, UK

Fundamenta Mathematicae, Tome 193 (2007) no. 1, pp. 37-77.

Résumé

We obtain estimates for derivative and cross-ratio distortion for $C^{2+\eta}$ (any $\eta>0$) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition.For two intervals $J \subset T$, the cross-ratio is defined as the value $$B(T,J):=\frac{|T|\,|J|}{|L|\,|R|}$$ where $L,R$ are the left and right connected components of $T\setminus J$ respectively. For an interval map $g$ such that $g_T:T \to \mathbb R$ is a diffeomorphism, we consider the cross-ratio distortion to be $$B(g,T, J):=\frac{B(g(T),g(J))}{B(T,J)}.$$We prove that for all $0 K 1$ there exists some interval $I_0$ around the critical point such that for any intervals $J \subset T$, if $f^n|_T$ is a diffeomorphism and $f^n(T) \subset I_0$ then $$B(f^n, T, J)> K.$$ Then the distortion of derivatives of $f^n|_J$ can be estimated with the Koebe lemma in terms of $K$ and $B(f^n(T),f^n(J))$. This tool is commonly used to study topological, geometric and ergodic properties of $f$. Our result extends one of Kozlovski.

Détail

DOI : 10.4064/fm193-1-4

Keywords: obtain estimates derivative cross ratio distortion eta eta unimodal maps non flat critical points require schwarzian like condition intervals subset cross ratio defined value frac where right connected components setminus respectively interval map mathbb diffeomorphism consider cross ratio distortion frac prove there exists interval around critical point intervals subset diffeomorphism subset distortion derivatives estimated koebe lemma terms t tool commonly study topological geometric ergodic properties result extends kozlovski