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Journal of the American Mathematical Society

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Real bounds, ergodicity and negative Schwarzian for multimodal maps


Authors: Sebastian van Strien and Edson Vargas
Journal: J. Amer. Math. Soc. 17 (2004), 749-782
MSC (2000): Primary 37Exx, 37Fxx
DOI: https://doi.org/10.1090/S0894-0347-04-00463-1
Published electronically: August 27, 2004
Erratum: J. Amer. Math. Soc. 20 (2007), 267-268.
MathSciNet review: 2083467
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Abstract: We consider smooth multimodal maps which have finitely many non-flat critical points. We prove the existence of real bounds. From this we obtain a new proof for the non-existence of wandering intervals, derive extremely useful improved Koebe principles, show that high iterates have ‘negative Schwarzian derivative’ and give results on ergodic properties of the map. One of the main complications in the proofs is that we allow $f$ to have inflection points.


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Additional Information

Sebastian van Strien
Affiliation: Department of Mathematics, Warwick University, Coventry CV4 7AL, England
Email: strien@maths.warwick.ac.uk

Edson Vargas
Affiliation: Department of Mathematics, University of São Paulo, São Paulo, Brazil
Email: vargas@ime.usp.br

Keywords: Dynamical systems, interval dynamics, holomorphic dynamics
Received by editor(s): May 1, 2002
Published electronically: August 27, 2004
Additional Notes: The first author was partially supported by EPSRC grant GR/R73171/01.
The second author was partially supported by CNPq-Brasil, Grant #300557/89-2(RN)
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.