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

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

2010-05-26

arxiv.org/abs/1005.4828v1

Mathematics

Dynamical Systems

44 pages

Scientific paper

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.

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