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Abstract

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.