Distortion bounds for C2+η unimodal maps

Mike Todd*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 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.

Original languageEnglish
Pages (from-to)37-77
Number of pages41
JournalFundamenta Mathematicae
Issue number1
Publication statusPublished - 2007


  • Cross-ratio distortion
  • Unimodal map


Dive into the research topics of 'Distortion bounds for C2+η unimodal maps'. Together they form a unique fingerprint.

Cite this