Real bounds, ergodicity and negative Schwarzian for multimodal maps

van Strien, Sebastian; Vargas, Edson

doi:10.1090/S0894-0347-04-00463-1

Real bounds, ergodicity and negative Schwarzian for multimodal maps
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by Sebastian van Strien and Edson Vargas;

J. Amer. Math. Soc. 17 (2004), 749-782

DOI: https://doi.org/10.1090/S0894-0347-04-00463-1

Published electronically: August 27, 2004

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Erratum: J. Amer. Math. Soc. 20 (2007), 267-268.

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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