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

Abstract

We obtain estimates for derivative and cross--ratio distortion for $C^{2+η}$ (any $η>0$) 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 $g$ \st $g_T:T \to \RRR$ 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.$$ 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 $f$. This extends a result of Kozlovski.