## 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 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 I_{0} around the critical point such that for any intervals J ⊂ T, if f^{n}|_{T} is a diffeomorphism and f^{n}(T) ⊂ 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.

Original language | English |
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Pages (from-to) | 37-77 |

Number of pages | 41 |

Journal | Fundamenta Mathematicae |

Volume | 193 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 |

## Keywords

- Cross-ratio distortion
- Unimodal map

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