We obtain estimates for derivative and cross-ratio distortion for (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 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 is a diffeomorphism and fⁿ(T) ⊂ I₀ then B(fⁿ,T,J) > K. Then the distortion of derivatives of 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-1-4,
author = {Mike Todd},
title = {Distortion bounds for $C^{2+$\eta$}$ unimodal maps},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {37-77},
zbl = {1114.37020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-1-4}
}
Mike Todd. Distortion bounds for $C^{2+η}$ unimodal maps. Fundamenta Mathematicae, Tome 193 (2007) pp. 37-77. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-1-4/