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

Mike Todd. "Distortion bounds for $C^{2+η}$ unimodal maps." Fundamenta Mathematicae 193.1 (2007): 37-77. <http://eudml.org/doc/283323>.

@article{MikeTodd2007,
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 ⊂ 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 $g_T: T → ℝ$ 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 $fⁿ|_J$ 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.},
author = {Mike Todd},
journal = {Fundamenta Mathematicae},
keywords = {cross-ratio distortion; interval map; critical point; Koebe lemma; topological, geometric and ergodic properties},
language = {eng},
number = {1},
pages = {37-77},
title = {Distortion bounds for $C^\{2+η\}$ unimodal maps},
url = {http://eudml.org/doc/283323},
volume = {193},
year = {2007},
}

TY - JOUR
AU - Mike Todd
TI - Distortion bounds for $C^{2+η}$ unimodal maps
JO - Fundamenta Mathematicae
PY - 2007
VL - 193
IS - 1
SP - 37
EP - 77
AB - 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 $g_T: T → ℝ$ 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 $fⁿ|_J$ 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.
LA - eng
KW - cross-ratio distortion; interval map; critical point; Koebe lemma; topological, geometric and ergodic properties
UR - http://eudml.org/doc/283323
ER -