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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.
@article{PMIHES_2011__114__171_0,
author = {Avila, Artur and Lyubich, Mikhail},
title = {The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {171--223},
publisher = {Springer-Verlag},
volume = {114},
year = {2011},
doi = {10.1007/s10240-011-0034-2},
zbl = {1286.37047},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-011-0034-2/}
}
TY - JOUR AU - Avila, Artur AU - Lyubich, Mikhail TI - The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes JO - Publications Mathématiques de l'IHÉS PY - 2011 SP - 171 EP - 223 VL - 114 PB - Springer-Verlag UR - http://geodesic.mathdoc.fr/articles/10.1007/s10240-011-0034-2/ DO - 10.1007/s10240-011-0034-2 LA - en ID - PMIHES_2011__114__171_0 ER -
%0 Journal Article %A Avila, Artur %A Lyubich, Mikhail %T The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes %J Publications Mathématiques de l'IHÉS %D 2011 %P 171-223 %V 114 %I Springer-Verlag %U http://geodesic.mathdoc.fr/articles/10.1007/s10240-011-0034-2/ %R 10.1007/s10240-011-0034-2 %G en %F PMIHES_2011__114__171_0
Avila, Artur; Lyubich, Mikhail. The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes. Publications Mathématiques de l'IHÉS, Tome 114 (2011), pp. 171-223. doi: 10.1007/s10240-011-0034-2
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