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

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.