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Summary: THE FULL RENORMALIZATION HORSESHOE
FOR UNIMODAL MAPS OF HIGHER DEGREE:
EXPONENTIAL CONTRACTION ALONG HYBRID CLASSES
ARTUR AVILA AND MIKHAIL LYUBICH
Abstract. We prove exponential contraction of renormalization along hybrid
classes of infinitely renormalizable unimodal maps (with arbitrary combina-
torics), 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 ab-
stract 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.
Contents
1. Introduction 1
2. Hybrid classes, external maps, and renormalization 5
3. Path holomorphic spaces, the Carath´eodory metric and the Schwarz
Lemma 17
4. Hybrid leaves as Carath´eodory hyperbolic spaces 19
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