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## Fundamenta Mathematicae

2007 | 193 | 1 | 37-77
Tytuł artykułu

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

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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.
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Tom
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37-77
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wydano
2007
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• Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK
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