We extend the renormalization operator introduced in [3] from period-doubling Hénon-like maps to Hénon-like maps with arbitrary stationary combinatorics. We show the renormalisation prodcudure also holds in this case if the maps are taken to be strongly dissipative. We study infinitely renormalizable maps F and show they have an invariant Cantor set O on which F acts like a p-adic adding machine for some p > 1. We then show, as for the period-doubling case in [3], the sequence of renormalisations have a universal form, but the invariant Cantor set O is non-rigid. We also show O cannot possess a continuous invariant line field.
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.