Publications mathématiques de l'IHÉS

, Volume 114, Issue 1, pp 171–223 | Cite as

The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes

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Abstract

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree d. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator (“beau bounds”), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.

Keywords

Schwarz Lemma Holomorphic Motion Renormalization Operator Hybrid Lamination Beltrami Differential 
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Copyright information

© IHES and Springer-Verlag 2011

Authors and Affiliations

  1. 1.CNRS UMR 7586Institut de Mathématiques de JussieuParisFrance
  2. 2.IMPARio de JaneiroBrazil
  3. 3.Department of MathematicsStony Brook UniversityStony BrookUSA

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