Higher order Schwarzian derivatives in interval dynamics

Volume 206 / 2009

O. Kozlovski, D. Sands Fundamenta Mathematicae 206 (2009), 217-239 MSC: 37E05, 53A55, 41A20, 30C20. DOI: 10.4064/fm206-0-12


We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up to any given order.


  • O. KozlovskiMathematics Institute
    University of Warwick
    Coventry CV4 7AL, United Kingdom
  • D. SandsCNRS, Département de Mathématiques
    Université Paris-Sud
    91405 Orsay Cedex, France

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