Institute for Mathematical Sciences
Artur Avila, Mikhail Lyubich
The full renormalization horseshoe for unimodel maps of higher degree: exponential contraction along hybrid classes
Abstract: We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodel 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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