Distortion bounds for C^2+η unimodal maps

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 ⊂ T, the cross-ratio is defined as the value B(T,J) := |T||J|/|L||R| where L, R are the left and right connected components of T \ J respectively. For an interval map g such that gT : T → ℝ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J) := B(g(T),g(J))/B(T,J) We prove that for all 0 < K < 1 there exists some interval I₀ around the critical point such that for any intervals J ⊂ T, if fⁿ|_T is a diffeomorphism and fⁿ(T) ⊂ I₀ then B(fⁿ,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(fⁿ(T), fⁿ(J)). This tool is commonly used to study topological, geometric and ergodic properties of f. Our result extends one of Kozlovski.